Collected Papers: Volume I 1955-1966

--t

9

be the isomorphism defined by letting
£

H and let ~r(B(f)) =

a

and ~,r(B'(f)) =

a';

then

= a'.

<pr(a)

Proof: If u

£

Hr(g') then clearly

(a,
~r(U)

=

(B(t) , ~r(U)).

is contained in the subspace ru,. Hence

(B(!) , ~r(u))

=

(B'(f) , ~r(U))

Thus (a,
= (a', u). Q.E.D.

87

261

ON THEOREMS OF FROBENIUS AND AMITSUR-LEVITSKI

Since, among other things, we are in the business of proving equivalences we will not yet assume knowledge of the FROBENIUS theorem. From §4.7 we know, without the FROBENIUS theorem, that the elements h~ I: IfJ defined by _ 1_ C (1) hfJo -_ v' Jl8 XfJ

-

XfJ<-I»)

C5.4.1)

form a basis of H. hg

Now let RH be the kernel of B. It is immediately clear from Lemma 4.8 that £ nH if and only if l({3) > n and that, in fact, all such hg form a basis of RH. We now have the following theorem on primitivity. Theorem 5.4. Assume r

=

1, p

2p -

br

>

1. Let b r I: Hr(g) be defined by

~r(B(h~))

=

where E = e(r) is given by (4.8.2) and h~ is defined as above. Then b r I: b r vanishes if and only if p > n.

reg) and

Proof: Since lee) = p it follows from (5.4.1) h~ I: nH if and only if p > n. Hence b = 0 if and only if p > n. Thus we may assume p ~ n. Let m = p - 1 and let U' be chosen as in §5.4. Then h~ I: mH and hence from Lemma 5.4 T

",rCb

T)

= O.

Now observe the uniqueness of E = fer) in the following regard. If s ~ r and (3 = (3(s) is any self-associated partition of 8, l({3) ~ p where the equality sign holds if and only if s = rand {3 = E. Thus if (3 = (3(s) =1= e(r) then l({3) ~ m and hence hg 1- mH" . Consequently by Lemma 5.4 ","CaB)

where a" kernel of br

=1= 0

In other words, the kernel of "," = 0 for 8 < r and the is (b r ). We now apply the criterion Lemma 2.6 and conclude that

= ~(B(hm. ",r

I: rCg). Q.E.D. 5. Having obtained Theorem 5.4 we now observe that the FROBENIUS theorem

immediately yields the formula of DYNKIN. Indeed, the FROBENIUS theorem asserts h~ = h, . But then Theorem 5.4 asserts that the element br I: r(g) defined by (b r ,u)

= (7jT(U) , B(h,))

(5.5.1)

vanishes if and only if p > n, where r = 2p - 1, and for p < n is the invariant cocycle in the primitive class b r. But now if u = Al 1\ ... 1\ Ar then by (4.9.3) (b r ,AI 1\ A2 1\ ... 1\ Ar) = c·tr [AIA2 ... AT]

for a non-zero scalar c. This (up to scalar multiple) is the formula of DYNKIN. A second consequence of an application of the FROBENIUS theorem is the

88

262

BERTRAM KOSTANT

following sharpening of a result of WEYL. In [5], pp. 235-237, WEYL shows only that if p > n the class b, corresponding to the multilinear form br defined by (5.5.1) depends upon the classes b. and their cup products, writing 8 = 2q - 1, q ~ n. Here we have already seen that b, = 0 if p > n. Until now we have been concerned with proving the equivalence of I, II, and III of §1.2. Here we will give a direct proof that h~ = h•. 6. Let p + q = r. Define S(p) C S by S(P)

{u(:s I(a) 1 ~ i

=

<j ~
(b) p

N ow if E is a vector space, E* its dual, a familiar fact that (c p

/\

c.

< u(j)}. < u(j)

p implies u(i) r implies u(i) Cp

(:

/\ p

E* and c. (: /\. E* then it is

,Xl /\ X2 /\ ••• /\ X r )

L:

sg (u)(cp ,Xu(l) /\ ... /\ Xu(p)}(c. ,Xu(p+l) ... Xu(r»'

(5.6.1)

UtS(p)

N ow let u (: S" , r (: S. and let bu (: reg) and by (: r(g) be defined by bu = if;P(B(hU» ,

Also let f (: Hr be defined by if;'(B(f» (B(f), Al

=

= bu

/\

br . Then from (5.6.1)

® A2 ® ... ® Ar)

L:

sg (p)(B(h

L:

sg (p)(B(u) ,

ptS(p) ptS(p)

U ),

Ap(l)

L:

® ... ® Ap(p»(B(h') , Ap(p+l) ® ... ® A p(,»

sg (v)Apv(l)

® ... ® Apv(p»

peS})

. (B(r) ,

L: sg (/L)Ap(p+p(I) ® ... ® Ap(p+p(.» "ESO'

=

L: sg (p)Ap(I) ® ... ® A p(,»

(B(rN) ,

ptS,.

where 6-, f f(i) = p

£

= u(j), 1 ~ j ~ p, and 6-(j) = j for j > p; P and f(j) = j for 1 ~ j ~ p. Thus f can be

Sr are defined by 6-(j)

+ r(j -

p) for j = haT.

chosen so that f Similarly let U u

(:

>

Jp(g) and

U r (:

J.(g) be such that

if;p(u u) = B(hj

and if;.(u r) = B(hr).

Then since '1'(L: sg (p)(Ap(I) ptl>

where at = '1\A;) as above

£

® ... ® Ap(r») =

al

/\

•••

/\

ar

C1 (g), i = 1,2, ... , r, (see (5.2.2» by the same argument

89

263

ON THEOREMS OF FROBENIUS AND AMITSUR-LEVITSKI

y,.r(U

g

/\

p!q! aT Ur) = - , B(h ). r.

N OW recalling the role played by A as given in §4.6 we obtain the following information on cup and Pontrjagin products. Lemma 5.6. Let p H.(g) be such that

+

q = r. Let fi1

t\r,,«U,,)) and let a"

E

H"(g) and a.

E

E

no(p) and fi2

t\r.«U.))

B«h{J,)) ,

=

=

E

no(q). Let u"

E

H,,(g),

U. E

B«h{J,))

H·(g) be such that

t\r"(B«h{J,)))

= (a.,) ,

tl!·(B«h{J,))) = (a.).

+

Let 0 = o(fi1) O(fi2) E nCr) be the partition sums. Then by §4.6 if 0 f. A(r), V U. = 0 and a" V a. = O. On the other hand, if 0 E A(r) and fi E no is such that S(fJ) = 0 then

Up

and tl!r(B«h{J))) = (a" V a.).

Now let r = 2p - 1 where p ~ n. Let fi E no and let U E Hr(g) be such that tl!r(u) = B(h{J). Then if fi =f= E it follows immediately from Lemma 5.6 that U E Dr(g). On the other hand, we have already proved (Theorem 5.4) tl!r(B(h~))

E

r(g).

Thus we have the following orthogonality relation: (B(h~), B(h{J))

= 0

(5.6.1)

for all fi E no , where fi =f= E. Now observe from (5.4.1)

(h~, . hg,) =

{O

if fi1 =f= {32 if {31 = {32 •

(e{J,)

Hence upon writing (see §4.7) hfJ = LfJ' a{J,h~, , it follows that (h{J' h~)

=

a,(e,).

Since tr B(h{J' h~)

(B(h{J) , B(h~))

=

it follows then from (5.6.1) that a, = 0 since tr B(e,) =f= O. Thus if {f, g} designates the usual inner product (in 1),

If,

g}

=

L

ati

90

f(q)g(q)

264

BERTRAM KOSTANT

for f, gEl, it follows that {hp, h~} = 0 if {3 of the basis {h p }, h~

=

=1=

E.

But now expanding h~ in terms

L cphfJ ,

and observing that the hp are obviously mutually orthogonal in I it follows that 0 for all (3 =1= E. Hence

Cp =

h~ = c,h,

so that h, E I, . Thus we have proved h, E I, directly. If r = 2p - 1 and U r E Hr(g) is such that 'l7r(u r) = B(h,) note that Lemma 5.6 and the FROBENIUS theorem imply that U r E Pr(g). Finally, turning our attention to the general element {3 E go , Lemma 5.6 yields Theorem 5.6. Let (3 E goer) and let 01 , 02 , •.. , op be the lengths of the "shells" of the graph G({3) of {3 (see §4.5). If b p E Hr(g) is defined by b/l then bp vanishes if and only

if

01

>

=

,fCB(h/l))

2n - 1. Moreover,

(bp) = (b a, U b a, U ... U b 6.) where b a, is a primitive oi-cohomology class.

Clearly, the analogous homological statement is also true. BIBLIOGRAPHY 1. A. S. AMITSUR & J. LEVITSKI, Minimal identities for algegras, Proc. A mer. Math Soc. (1950), pp. 449-463. 2. J. L. KOSZUL, Homologie et cohomologie des algebres de Lie, Bull. Soc. math. de France 78 (1950), pp. 65-127. 3. H. BOERNER, Darstellungen von Gruppen, Springer, Berlin, 1955. 4. G. FROBENIUS, tJ"ber die Charaktere der alternierenden Gruppe, Sitzgsber. preuss. Akad. Wiss. 1901, pp.303-315. 5. H. WEYL, The Classical Groups, Princeton Univ. Press, Princeton, 1946. 6. N. JACOBSON, Structure oj Rings, Amer. Math. Soc. Colloquium Pub., v. 37. 7. E. B. DYNKIN, Topological characteristics of homomorphisms of compact Lie groups, Mat. Sb. N.S. 35 (77) (1954) pp. 129-173. 8. 1. SCHUR, tJ"ber die rationalen Darstellungen der allgemeinen linearen Gruppe. Sitzgsber. preuss. Akad. Berlin 1927, pp. 58--75. 9. H. CARTAN, La transgression dans un groupe de Lie et dans un espace fibre principal, Colloque de Topologie, Bruxelles, 1950, pp. 57-71.

University of California Berkeley, California

91

Reprinted from the DUKE MATHEMATICAL JOURNAL Vol. 25, No. 1, pp. 107-123 March, 1958

A CHARACTERIZATION OF THE CLASSICAL GROUPS By BERTRAM KOSTANT

1.1. Introduction. By one method of classification there are three types of (complex, connected) classical groups, (a) GL(n. C), (b) SO(n, C), and (c) Sp(n, C). So designated, each type is given as a specific group of matrices. It is perhaps neater (and for us more pertinent) to describe these groups by means of the special linear representation which each type admits. That is, if V is a complex vector space of dimension m, then up to an inner automorphism GL(m, C) is in a natural way isomorphic to the group GL(V) of all non-singular linear transformations on V. If B is a non-singular bilinear form on V, then the identity component of the subgroup of GL(V) which leaves B invariant is in a natural way (again up to an inner automorphism) isomorphic to SO(m, C) in case B is symmetric and to Sp(n, C) in case m = 2n and B is skew-symmetric. If G is a classical group, then the isomorphism 71": G ---7 GL(V) described above defines a representation of G which we shall call a natural representation. It is a simple property that a natural representation is irreducible and that any two such representations are equivalent. (Hence, of course, such a representation defines only one member, but a special member, of the infinite collection of equivalence classes of equivalent irreducible representations of G.) We shall call a subgroup GO, where GO C GL(V), a classical linear group if GO is isomorphic to a classical group G under a natural representation of G. 1.2. Now in general (actually m ~ 4) GL(V) contains many connected Lie subgroups GO, other than just classical linear ones, which act irreducibly on V. In this paper we shall characterize among all such groups those which are classical linear. The result, although simply stated in terms of the group GO itself, is even simpler to state in terms of the Lie algebra gO of G. In this case the characterization involves only the existence of a single non-nilpotent element of gO of sufficiently low rank. Our main result (Theorem 2.5) yields the following corollary: COROLLARY. Let gO be a complex Lie algebra of linear transformations acting irreducibly on the m-dimensional complex vector space V. (1) Then gO is the Lie algebra of all linear transformations on V if and only if it contains a non-nilpotent operator A of rank 1. (2) If gO leaves invariant the non-zero bilinear form B on V, then gO is the Lie algebra of all linear transformations leaving B invariant if and only if gO contains a non-nilpotent operator of rank 2. Remark 1. As usual the invariance of a bilinear form B under the action of a Lie algebra gO is taken to mean (Au, v) (u, Av) = 0 for all u, v I: V, A I: gO.

+

Received March 29, 1957; in revised form, October 28, 1957.

107 B. Kostant, Collected Papers, DOI 10.1007/b94535_9, © Bertram Kostant 2009

92

108

BERTRAM KOSTANT

Here (u, v) designates the value of B on the pair u, v E V. Throughout this paper we shall use a bold faced B to designate a non-singular bilinear form. Possibly, however, B will occur with a subscript (e.g. Bo). In such a case, however, its value on a pair of vectors u, v will be denoted by (u, v) with the corresponding subscript (e.g. (u, v)o). With regard to (2) it should be recalled that since gO acts irreducibly B is (a) necessarily non-singular and (b) necessarily either symmetric or skew-symmetric. Hence (2) implies gO is the Lie algebra of a classical linear group. It might also be remarked that in a sense (1) qualifies as a density theorem for complex Lie algebras. The condition, of course, is of a different nature than the corresponding theorem for associative algebras. The machinery used in the proof of Theorem 2.5 is the Cartan-Weyl theory of representations. H. Flanders has given a direct proof of (1) in the above corollary using only ideas in the linear algebra. It seems likely that (2) in the corollary can be proved in a similar way. Even so, it is felt that the use of the more elaborate weight theory is justified principally because viewed in the form of Theorem 2.5 the above corollary, rather than being an isolated fact, fits in neatly as a special case (defined by an orthogonality condition on the weights) in the larger framework of representation theory. 1.3. In §3 we give an application of the above corollary to a topic in differential geometry. More specifically, it is used to settle a question regarding curvature in Riemannian symmetric spaces. We consider the comparison of sectional curvature as computed on decomposable bivectors (planar elements) with the more general sectional curvature as computed on all bivectors (skewcontravariant tensors of degree 2) . We show that a necessary and sufficient condition that the more general sectional curvature take a non-zero maximum or minimum (or indeed any non-zero critical value) on a planar element is that the space have constant curvature.) 2.1. MAIN ThEOREM. Let 9 be a complex reductive Lie algebra. That is, g may be written, 9 = 3 EB 13 where sis the center of 9 and 13 = [g, gJ is semi-simple. Let g be a Cartan subalgebra of g. Then; as one knows, g may be written g = 3EB t where t = 13 (\ g is a Cartan subalgebra of 13. Let g* be the dual space of g. In a natural way g* decomposes as g* = 3* EB t* where 3* is orthogonal to t and t* is orthogonal to 3. AIl representations 7r of 9 considered here will be on a finite dimensional complex vector space V. They will be assumed to be complex homogeneous and completely reducible (that is, 7r(X) is diagonalizable for any X E z). Each representation 7r of 9 induces an invariant inner product B .. on g. We designate the value of B". on X, Y E 9 by (X, Y)".. The latter is defined as tr 7r(x)7r(Y). As one knows, the elements of 7r(g) are simultaneously diagonalizable. That is, there exists a basis Vi of V such that 7r(X)Vi = Ai(X)V i for all X E g. The Ai are elements of g* and are the weights of 7r. For each A E g* we designate by nX(7r) the dimension of the eigenspace for A in the representation

93

109

A CHARACTERIZATION OF THE CLASSICAL GROUPS

7r. That is, n).(7r) is the multiplicity of the weight X for 7r. We shall let .6(7r) be the set of non-zero weights of 7r. 2.2. In all our considerations we shall assume (as of course we may) that g arises as the complexification of the Lie algebra f of a compact Lie group K. (We do this simply to avoid complications involving the center of g. That is, the "real" and "imaginary" parts of ~ are now fixed. Thus 9 = r if. Now ~ may be chosen so as to be the complexification of the Lie algebra go of a maximal torus T C K. Thus also ~ = go + igo . In a natural way then we may write ~* = g~ + ~9~ where g~ is the real dual to the real linear space go . Similarly we shall consider only those representations 7r of g which arise from complex representations of K. Thus it is clear that if X I: .6(7r) , then X I: ~9~ since X must be pure imaginary on go. Moreover, if I C ~u is the discrete (Abelian) group defined by I = {X I: go I exp X = e} where e is the identity of K, then X I: 1* C ig~ where I* is defined by I* = (TJ I: ~9~ I TJ(X) = 2n7ri, X I: I for some integer n depending on Xl. The discrete (Abelian) group I* is, as one knows, the set of all weights of all considered representations of g. Independent of any representation, however, we shall refer to the elements of I* as weights. Now in the usual way we may impose a lexicographical ordering in if)~ • Hence aside from the zero element of if)~ any element is either positive or negative. We designate by .6 +(7r) (respectively .6 -(7r)) the positive (respectively negative) weights of 7r. 2.3. N ow if 7r is a faithful representation of g, then the restriction of B .. to ~ is non-singular (since it is negative definite on go). Thus there exists a natural isomorphism between f)* and f) when 7r is specified. We shall retain the convention of using lower case Greek letters for elements of g* and adopt the convention of using the corresponding upper case Greek letters with the superscript 7r for the associated elements of f). Thus is 7r is faithful and w I: ~* is given, then n" I: ~ is defined by w(X) = (n", X) .. for all X I:~. Furthermore the isomorphism w ~ n" permits us to introduce B .. in g*, and henceforth we shall understand that B". is so defined when 7r is faithful. It is clearly then positive definite on if)~ . N ow associated to every representation 7r of g on V is the contragredient representation 7r* of 9 on V*, the dual to V. One knows that .6(7r) = - .6(7r*); see e.g. [12]. The following lemma will be used later.

+

LEMMA 2.3. Let 7r be a faithful representation of 9 on V and let 7r* be the contragredient representation. In the usual way identify V ® V* with the set gr(v) of all operators on V so that we may regard 7r(g) C V ® V*. Let r: V ® V* ~ 7r(g) be the orthogonal projection of V ® V* on 7r(g); that is, orthogonal with respect to the inner product (A, B)o = tr AB for A, B I: V ® V*. Now if with respect to g, V). I: V and v'!\ I: V* are weight vectors for the weights X I: .6(7r) and - X I: .6(7r*), then

rev).

® v:!\) = (VA , v:~\) .7r(A '").

94

110

BERTRAM KOSTANT

Remark 2. With regard to the definition of r it should perhaps be remarked that although the inner product defined above is not positive or negative definite, it is still (1) non-singular on V ® V* and (2) may be used to define a complement to a subspace (e.g. 71'(g» whenever its restriction to the subspace is nonsingular (e.g. 71'(g». Also, as in the proof below, one may speak of operators on V ® V* (e.g. ad A) which are symmetric or skew-symmetric (e.g. ad A) with respect to the inner product. Proof. For any A £ V ® V* let ad A be the operator on V ® V* defined by ad A(B) = [A, B] where B £ V ® V*. It is a simple property of the inner product that ad A is skew-symmetric for all A £ V ® V*. But then if A £ 71'(g), ad A must leave the orthocomplement to 71'(g) invariant since it leaves 71'(g) invariant. That is, ad A commutes with r for all A £ 71'(g). Thus for any X £~, (7I'(X), r(v~

® v!~»o

=

(7I'(X) , v~ ® v!~)o

=

(7I'(X)v~

, v!~)

= h(X)(V~ , v!.).

But by definition of A", (7I'(X),7I'(A"»o = heX).

Thus r(v~ ® v!).) = (v. , v!.) 71'(A "), which concludes the proof. 2.4. Now define ~o = ~o (\ ~, ~~ = ~~ (\ ~*, to = ~o (\ t, t~ = ~~ (\ t*, so that ~o = ~o + to , ~~ = A~ + t~ • In case 71' is the adjoint representation (i.e. 71' = ad) we shall write B, (X, Y) and d instead of B" , (X, Y) .. and d(7I'). Of course d C it~ and since 71' is faithful on £I there is a natural correspondence between t and t*. We shall write

£

i~3'

I w(
~

0 for all cp £ d +} .

Since a corresponding set in i~o is known as the Weyl (closed) chamber we shall refer to D as the Weyl (closed) cochamber. Let Da = u(D) for U £ W. One obtains in this manner all the Weyl cochambers determined by all systems of positive roots. We recall some facts concerning these matters which we shall need and refer the reader to §4, [8]. (1) For any w £ ~1)* there exists U £ W such that uw £ D. (2) Concerning representations, if 71' is irreducible, then the highest weight h of 71' (which as one knows determines 71') is contained in D. Also n.(7I') = 1. (3) In any w £ 1*, U £ W, one has n.,(7I') = na.,(7I'). The following asserts that if 9 is simple the angle (with respect to B) between any two vectors in D is less than 90°.

95

A CHARACTERIZATION OF THE CLASSICAL GROUPS

LEMMA 2.4. we have (Il, II)

Assume g is simple (also non-Abelian).

Then for any Il,

111 II £

D

> O.

Proof. Let £¥l , ••• , £¥r be the simple positive roots. Let El , ••• , Er £ if)~ be defined by (E. , £¥j) = 8. j • The Ei form a basis in if)~ , and it is immediately clear that upon writing w = L:-l aiE, for any w £ ~1)~ ,

D = {w tiM I ai Thus it suffices to show that by definition of B

But if cp £ .:i +, then cp =

(E, , Ej)

~

0, i = 1,2, ... ,r}.

>

0 for all i, j = 1, 2, ... ,r. But now

L:=1

ni£¥i where the ni are non-negative integers. Thus O. Hence, to prove the lemma it suffices to show the existence of a root cp such that n,nj > O. But it is a well-known fact (see e.g. [2; 216]) that the highest root of a simple complex Lie algebra satisfies this condition. This completes the proof. Note. If 71" is any non-zero representation- of g, then we may clearly substitute B .. for B in Lemma 2.4 since B .. = aB where a > O. 2.5. Now we recall that the Lie algebra of SL(n, C) is written as A n - 1 so that the Lie algebra of Gl(n, C) is A n _) EB Dl where D) is the one-dimensional complex Lie algebra. Also the Lie algebras of SO(n, C) is Dn/2 in case n is even and B(n-ll/2 in case n is odd. Finally the Lie algebra of Sp(n, C) is C" ; the latter is, of course, a set of 2n X 2n matrices. Of course the notion of natural representation carries over to these classical Lie algebras. In the case of Lie algebras, however, the question of establishing whether a representation is natural or not becomes simpler. This is so mainly because the Cartan-Weyl theory of weights which is employed is more directly applicable to Lie algebras. The following theorem gives a simple criterion (an orthogonality condition on the weights) when given a representation of a Lie algebra g, that (1) g is isomorphic to a classica~ Lie algebra and (2) the representation is then equivalent to a natural one. (Ei

,cp) (cp,

Ej)

=

ninj

~

THEOREM 2.5. Let 71" be a faithfuL irreducible representation of a complex Lie algebra g on V. Let n = dim V. (1) Then (a) g is isomorphic to A n - 1 EB D) and (b) with respect to this isomorphism 71" is equivalent to the natural representation of A n - 1 EB D) if and only if there exists a weight II £ .:i(7I") which is orthogonal, with respect to B .. , to any Il £ .:i(7I") for which Il ~ II. (2) Then (a) g is isomorphic to B (,,-1)/2 in case n is odd, or either Dn/2 or C,,/2 in case n is even and (b) with respect to this isomorphism 71" is equivalent to the natural representation in any of these cases if and only if there exists a weight

96

112 11 I:

BERTRAM KOSTANT

~(11")

any

p.

I:

such that - 11 I: ~(11") and such that for which p. ¢ ± 11.

p

is orthogonal, with respect to B". , to

~(11")

Proof. It is simple fact (see e.g. [7; 52] translation) since 11" is faithful and irreducible that 9 is reductive. In the notation of §2.1 let :L!-l g' = 13 be the direct sum decomposition of 13 into its simple ideals. Write a = gO so that 9 = :L!=o g'. Then l) = :L!-o l)8 where l)8 = g' n l). Of course l)' is a Cartan subalgebra of g'. The notation of §§2.3-2.5 established relative to l) and 9 will carry the same meaning here relative to l)' and g' when it appears with the superscript s. Furthermore if X I: l) (respectively WI: l)*) then X' (respectively w') will designate its component in l)' (respectively l)*') so that X = :L!-o X· (respectively W = :L!=o w'). N ow since 11" is irreducible and faithful, we observe that gO = a is at most one-dimensional and that p.~ = p.~ for any P.l , P.2 I: ~(11"). Thus if X I: ~(11") is the highest weight of 11" and p. is any other weight, then X - p. I: it~. In fact we may write X - p. = :L~-l P;oi; where Oil , ... , Oi r are the simple positive roots and PI , ... , p; are non-negative integers. (See [13], Theoreme 1, p. 17--04). Now we recall the well-known fact (see e.g. [8; 376]) that if p. I: ~(11") and cp I: ~, then 2(p., cp)/(cp, cp) = p - q where p and q are the largest non-negative jcp I: ~(11") when - p ~ j ~ q. In particular p. cp I: ~(11") integers for which p. if (p., cp) < 0 and p. - cp I: ~(11") when (p., cp) > o. Let L". be the operator on l)* defined by (L"Wl , W2) .. = (WI, W2). Since there is only one invariant bilinear form, up to a scalar multiple on a simple Lie algebra (see [11; Theoremes 11.1 and 11.2]), it follows for all s that the l)*' are eigenspaces for L". Moreover, if c. is the corresponding eigenvalue, c. > 0 for s ~ 1, and Co = o. Since any cp I: ~ is contained in il)~' for some s, it follows that 2(p., cp) / (cp, cp) = 2(p., cp),,/(cp, cp) .. so that we may replace B". for B in the criterion above determining whether p. + cp or p. - cp is contained in ~(11"). Consider the decomposition X = :L!-o X' of the highest weight X. The condition that 11" is faithful is equivalent to the condition that X' ¢ 0 for all s. To see this we first observe a ¢ 0 immediately implies that X° ¢ O. Now if X' = 0 for s ~ 1, then from the criterion above 1I"(E ,,)vA = 0 whenever cp I: ~. = ~ n il)*', E" is the corresponding root vector and VA I: V is the weight vector for X. Indeed since p = q and X ± cp cannot both be weights, p = q = O. Hence 1I"(g")vA = o. Since g' is an ideal in 9 and 11" is irreducible, this implies 1I"(g') = O. But this contradicts the faithfulness of 11". Now if p I: ~(11") is orthogonal to all other elements of ~(11"), this is obviously true of UP for any u I: W. Hence we may assume P I: D. We now show that P = X. Since (il)*", '/,1)*') = 0 for s ¢ t, it follows that

+

+

k

1

k

(p, X) .. = :L (p", X') .. = (pO, XO) .. 8=0

+ :L C,a- (p', X'). 1

Since .,,0 = XO, it then follows from Lemma 2.4 that (.,,', X') ~ 0 for any s ~ 0, and (.,,', X') = 0 implies .,,' = O. Hence (.", X) .. > O. But by our hypothesis

97

A CHARACTERIZATION OF THE CLASSICAL GROUPS

113

this can only happen if A = II. Now for any Jl t Ll(7r) there exists U t W such that UJl t D. By a similar argument then it follows that UJl = A. Thus we see that one obtains all occurring non-zero weights by applying W to A. Moreover, since n,,(7r) = n.,,(7r), they all occur with multiplicity one. Let A = AI, A2 , ... , Ap be the elements of Ll(7r). Since Ai = UAI for some u, it follows that (Ai, A;),.. = 0 for i r!' j. But then, being mutually orthogonal, they are linearly independent. Hence P :::; r where r is the rank (dim f)) of g. On the other hand, were X t f) orthogonal to all Ai , it would follow that 7r(X) = O. But that contradicts faithfulness. Thus the Ai span if)*, and hence we conclude r = p. We have of course assumed condition (1) of the theorem. We observe in this case that 3 r!' O. Indeed, in general D n - D = if~ and hence if g were semi-simple, D n - D = O. On the other hand - D is a cochamber so that there exists A' t - D n Ll(7r). But then A r!' A'SO that (A, A') .. = O. But also - A' t D so that CA, -A') .. =

t

8=1

1. (A',

_A'B) = O.

Ca

This, however, contradicts Lemma 2.4 since A' r!' 0 for all 8 ;;::: 1. On the other hand, if we assume (2) then 3 = O. This is immediate since as we have observed 11° = Jlo for any Jl t Ll(7r). But since -II t Ll(7r) , 11° = - 11° = 0, and hence 3 = 0 by the faithfulness of 7r. Now the property assumed for II in (2) is clearly invariant under the action for W. Hence as above we may assume II t D. In an argument identical with that of the previous case we may also conclude that A = II. Furthermore any Jl t Ll(7r) is of the form UA for some U t W so that n,,(7r) = 1. It also follows then that - Jl t Ll(7r) so that Ll(7r) = - Ll(7r). (This latter fact is, as one knows, equivalent to the condition that 7C" be self-contragredient; see [12] for example.) Now let AI, ... , Ap be the elements of Ll+(7r) where A = AI' Then, as above, (Ai , A;) .. = 0 for i r!' j and as before, we conclude that P = r (but in this case P is half the number of non-zero weights). We have deduced no information as yet about the zero weight. Write, as a direct sum V = VO + V' where VO is the weight space for the zero weight and Viis the subspace spanned by the weight vectors for the elements of Ll(7r). We have shown under the assumptions of (1) that dim V' = r and for (2) dim V' = 2r. On the other hand, for (1) it is immediate that VO = 0 since for any weight Jl, Jlo = AO r!' O. Hence in that case dim V = n = r. We now show that for (2), VO is at most one-dimensional. Assume VO r!' O. Then A= L~~l Piai where the Pi are positive integers (positive, by virtue of Lemma 2.4, and integral since the zero weight occurs) and the ai are the simple positive roots. Thus L:~l Pi = 8;;::: r. Now if E a , t g are the root vectors corresponding to the ai , and 'Y1 , ... ,'Y. are the roots - a; in some order, where - ai occurs with multiplicity Pi , then the vectors u, = 7C"(E'Yd,)7r(E'Yd') ... 7r(E'Yd.)VA ,

where T runs through all permutations of the integers from 1 to 8, span VO. Since VO r!' 0, at least one of the u, is not zero. We may assume this is the case

98

114

BERTRAM KOSTANT

for the identity permutation. Let 0; = - L.:;-1 'Yi and let V; = 7r(E"i+,)7r (E"i+,) ... 7r(E.,Jv),. Then Vi is a weight vector for 0; (: A+ (7r) , j = 1, 2, ... , 8. However, since Oi - 0; > 0 if i > j, the 0; are distinct. Thus (ai' 0;) ~ = 0 for i ~ j and hence the 0; are linearly independent in ~1)*. But since 8 ~ r, it follows in fact that 8 = r and that the 0; are identical with the Ai , in some order. We may assume Ai = Oi. Also since Pi = 1, we may assume 'Yi = - ai so that A; = 2::-1 ai , 1 :::; j :::; r. N ow consider U r where r is not the identity permutation. Let 1 < P :::; r be the greatest integer such that rep) ~ p. Let w = 7r(E_ a ,c.Jvp . Then w (: V'is a weight vector for the weight op - ar(p) = fJ. > o. But fJ. ~ Ai for i = 1, 2, ... , r and hence n,,(7r) = O. Thus w = 0 and hence O U r = o. It follows then that dim V = 1. We now prove under the assumptions of (1) (respectively (2» that if fJ. (: A(7r) (respectively if fJ. is a non-negative weight), then A - fJ. is a root, assuming, of course, that A ~ fJ.. We first observe that there exist positive roots f3i , i = 1,2, ... , q, such that A - fJ. = L.:~-1 f3i and such that v" = 7r),(E_ fl ,) 7r),(E-fl') ... 7r(E_ fJ ,)v), is non-zero and hence is a weight vector for fJ.. Let fJ.o = fJ., fJ.; = fJ. + L.:;-1 f3i· Then for all j, fJ.; is a weight of 7r and since fJ.i+l > fJ.i , all the fJ.i are distinct. Furthermore (fJ.i , fJ.i) or = O. This follows from distinctness in the case of (1) and distinctness together with the fact that fJ.i > fJ.o ~ 0 for i ~ 1 in the case of (2). Now it is obvious that fJ.l - fJ.o = f3 is a root. Assume we have shown fJ.j - fJ.o is a root. Then fJ.;+1 -

But for j

>

fJ.o =

(fJ.;+1 -

fJ.i)

+ (Mi

-

fJ.o) =

f3i

+ (fJ.i

-

fJ.o).

0

+

Thus f3; fJ.i - fJ.o = fJ.i+J -fJ.o is a root. Hence A - fJ. is also a root. Hence under the assumptions of (1), Ai - A; (: A for j = 2, ... r. Under the action of W this implies Ai - Ai (: A for all i ~ j. On the other hand, the Ai being linearly independent, this implies the roots obtained as differences in this way are distiI].ct. Thus the roots are at least r(r - 1) in number and hence dim g ~ r2. But dim V = n = r and since 7r is faithful, g must be isomorphic to grey), and 7r expresses an isomorphism. Hence of course, there exists an isomorphism between g and A n - 1 EB Dl such that the representation 7r corresponds to the natural representation of A n - 1 EB Dl . It may be recalled that there is an isomorphism of A n - 1 EB DJ on grey), where dim V = n, which is not equivalent (n > 2) to the natural representation. This representation is equivalent to the representation of A n - 1 EB Dl on An - 1 V which is induced by the natural representation of A n - 1 EB Dl on V. This, however, does not affect our result since it is only an isomorphism of g with A .. _l EB Dl with which we are concerned. We may restrict our consideration now to the assumptions of (2) . We have

99

A CHARACTERIZATION OF THE CLASSICAL GROUPS

115

shown A - p. £ .1 if p. is any non-negative weight distinct from A. N ow if is another non-negative weight, distinct from A and p., then (A -

p.,

A - p.')"

=

(A, A) ..

>

p.'

O.

Thus (A - p.') - (A - p.) = p. - p.' is also a root. If the zero weight occurs and we let p. or p.' equal zero, it becomes clear then that .1(11") C.1. On the other hand, given just that p., p.' £ .1(11") and that p. and p.' are linearly independent, we may choose the lexicographical order in such a way that p. and p.' are both positive (for example, by embedding p. and p.' in the order defining basis). Hence p. p.' £.1. Thus if Al , A2 , ... , Ar are the elements of .1+(11"), we see that ± Ai ± Ai , for i < j, are roots. Moreover, by the independence of the Ai we are thus assured of obtaining 2r(r - 1) roots in this manner. Also, if the zero weight occurs, then we obtain 2r more roots; namely ± Ai' Conversely, of course, if some Ai is a root, then the zero weight occurs for 11" since (Ai' Aj) .. > 0, and hence Aj - Aj = 0 is a weight. N ow consider the question as to whether we have exhausted all the possibilities for a root. If!p is a root, then since 1I"(E",) ~ 0 there must exist weights p., p.' such that p. - p.' =!p. Hence the only other possibility is that !p is of the form ± 2 Aj. Now we have seen that dim VO = 1 or 0 so that dim V = 2r + 1 or 2r according as the zero weight occurs or not. Case 1. dim V = 2r + 1. In this case ± Aj , j = 1, 2, ... , r, are roots. This excludes the possibility that ± 2Aj is a root for any j. Hence the totality of roots is given by the set ± Aj , ± Ai ± Ai for i < j. On the other hand, the linear independence of the Aj insures that these expressions for the elements of .1 in terms of the elements of .1(11") are unique. Hence the additive structure of .1 is given completely by our being able to write the roots in this way. But then, as one knows, .1 with the additive property so given, is isomorphic to root structure of Br (see e.g. [9; 126]) and hence g is isomorphic to B r . Moreover, the expression for the highest weight, Al , of 11" in terms of the roots, agrees with the expression of the highest weight of the natural representation of Br in terms of the roots of B r • Hence with respect to this isomorphism 11" is equivalent to the natural representation of Br = B en - ll /2 • Case 2. dim V = 2r. Recalling the transitive action of the Weyl group on the set {± Ai} we see that in this case it is simply a question as to whether all the ± 2Aj occur as roots or none occur. Both possibilities are realized. Indeed if the ± 2Ai are not roots, then ± Ai ± Ai for i < j defines (see argument above) the root structure of Dr , r ~ 2, (see e.g. [9; 126]) and if the ± 2Aj occur as roots, then ± Ai ± Ai , ± 2Ai , i < j, defines the root structure of Cr , r ~ 1. In either case the highest weight Al defines the natural representation. This concludes the proof. 2.6. Now we shall obtain the corollary of the introduction (Corollary 2.6) as a consequence of Theorem 2.5. The device enabling us to go from the orthogonality condition on a weight of Theorem 2.5 to the low rank condition of Corollary 2.6 is Lemma 2.3.

100

116

BERTRAM KOSTANT

COROLLARY 2.6. Let 9 be any complex Lie algebra of linear transformations acting irreducibly on the n-dimensional complex vector space V: (1) Then 9 is the Lie algebra of all linear transformations in V if and only if it contains an operator A of rank 1 such that A 2 ~ (i.e., tr A ~ 0). (2) If 9 leaves invariant (see Remark 1) the non-zero bilinear form B on V, then 9 is the Lie algebra of all linear transformations leaving B invariant if and only if 9 contains an operator A of rank 2 such that A 2 ~ 0.

°

Proof. As mentioned above, the fact that 9 acts irreducibly implies that 9 is reductive. Now, as in Lemma 2.3 identify V @ V* with the set gf(V) of all operators on V. N ow if an element A £ V @ V* is semi-simple as an operator on V (that is, A is diagonalizable) then, clearly, ad A as an operator on V @ V* is semisimple. If, furthermore, A £ g, then the restriction of ad A to 9 is semi-simple. That is, A is, by definition, a semi-simple element of g. It is well-known, however, that any semi-simple element of a reductive Lie algebra can be embedded in a Cartan subalgebra (see [9; 119]). Hence if A £ 9 is semi-simple as an operator on V, then it can be embedded in a Cartan sub algebra 1) of g. Now if A £ V @ V* is of rank 1, then A is of the form v @ v* for v £ V and v* £ V*. Clearly tr A = (v, v*). Hence if A 2 = (v, v*)A is not 0, then v is an eigenvector of A belonging to the non-zero eigenvalue, tr A. It follows immediately then that if A 2 ~ 0, A is a semi-simple operator on V. Hence A may be embedded in a Cartan sub algebra 1) of g. Let 7r designate the given representation of 9 on V. By commutativity it follows that v is a simultaneous eigenvector for any B £ 1), and hence v is a weight vector for a non-zero weight A £ .1(11} Furthermore, since [B, A] = 0, and since [B, A]

=

[B,v@v*]

=

Bv@v* +v@B*v*

=

A(B)v @ v*

+ v @ B*v*

= 0, it follows that v* is weight vector for the weight - A of the contragredient representation 7r* of g. Thus A = Vx @ v!x. On the other hand, since A has only one non-zero eigenvalue it follows that AJ (A) = for any Al £ .1C7r) for which Al ~ A. But now, by Lemma 2.3,

°

rCA) = r(vx @ v!x) =

(vx , v!x)A"

=

tr A·A".

Since A £ g, however, rCA) = A so that A = tr A· A". But then, if Al £ .1(7r) and AJ ~ A,

(Al , A) ..

=

(A; , A")"

101

A CHARACTERIZATION OF THE CLASSICAL GROUPS

=

t/A 1

= tr A =

(A~

,

117

A) ..

AI(A)

0.

Thus the conditions of (1), Theorem 2.5, are satisfied and hence (1) of Corollary 2.6 has been proved. N ow assume the condition of (2) in Corollary 2.6. Since g acts irreducibly, it follows that B must be non-singular. Moreover, again using the irreducibility, B is either symmetric or skew-symmetric. In any case'll" is self-contragredient so that among other things tr B = for any B t g. But then it follows easily that since rank A = 2 and A2 ;e 0, that A has 2 non-zero eigenvalues a and -a. Consequently A is semi-simple and hence may be embedded in a Cartan subalgebra l) of g. Let VI and V_I be the respective eigenvectors of A corresponding to the eigenvalues a and - a. As before it follows that VI and V-I are weight vectors for weights Al and A-I of'll". But .1('11") = - .1('11") and hence - Al t .1('11"). It follows then that A_I = - Al since A has rank 2 and A-I(A) = -Xl (A). Let X = AI' Now for any Btl), (A, B)" = tr AB = 2X(A) X(B). But then A" = (1/2X(A))A. On the other hand, in' t .1('11") and X' ;e ± X, then X'(A) = since A has rank 2. Using an argument identical to that above we conclude that (A', X) .. = 0, and hence the conditions of (2) in Theorem 2.5 are satisfied. The remainder of Corollary 2.6 follows immediately from the definitions of the natural representations of the classical Lie algebras, and in the case of Br , Cr and Dr , from the fact that the invariant form under such a representation is unique up to scalar multiple. 2.7. For the situation when we are dealing with the real numbers rather than the complex numbers we have the following corollary. Corollary 2.7 is actually weaker than Corollary 2.6 but it is more directly applicable for us (see §3.1).

°

°

COROLLARY 2.7. Let Vo be a real finite dimensional space and let B be a positive definite bilinear form on V. Let go be any Lie algebra of skew-symmetric (with respect to B) operators acting irreducibly on Vo. Then if go contains an operator W of rank 2, either (1) go is the Lie algebra of all skew-symmetric operators in Vo , or (2) Vo is even-dimensional, and there exists an operator J on Vo such that (a) J is skew-symmetric with respect to B, (b) r = - 1, (c) J commutes with every element of go. Moreover, if Vo is regarded as a complex vector space where i acts like J and C is the Hermitian inner product on Vo defined by

C(u, v)

=

(u, v)

+ i(u, Jv),

then go regarded as complex linear transformations on Vo is the Lie algebra of all skew-Hermitian operators in Vo .

102

118

BERTRAM KOSTANT

Proof. Let V = Vo + iVo be the complexification of Vo and g = go + ig o the complexification of go. We can of course regard g as acting on V. Case 1. Assume g acts irreducibly on V. Extend B to V (unique) as a complex bilinear form. It is clear of course that g leaves B invariant. Now W is still of rank 2 regarded as a complex operator on V. On the other hand W 2 :;c 0, since skew-symmetric operators are not nilpotent. Hence by Corollary 2.6 g is the Lie algebra of all operators on V which leaves B invariant. But then go must be the Lie algebra of all skew-symmetric operators on Vo . Case 2. V is not irreducible under g. In this case, as one knows, the ring of all operators on Vo which commute with g is isomorphic either to the complex numbers or to the quaternions. In either case we may find J, such that J2 = - 1, in the commuting ring of go. Since D = J + J* is symmetric and since D commutes with go , it follows that D = }..] for some scalar}... But tr J = 0 implies tr D = 0 and hence D = o. Thus J is skew-symmetric. Now let Vo be regarded as a complex vector space where i acts like J. It is clear that C is a Hermitian positive definite inner product and that go , with respect to C, is a Lie algebra of skew-Hermitian operators. Now if we identify g = go + ig o with go + J go , it follows that g is a complex Lie algebra of linear transformations on Vo which acts irreducibly. Since W has rank 1 with respect to the complex structure on Vo , it follows from Corollary 2.6 (since W 2 :;c 0) that g must be the Lie algebra of all complex linear transformations in Vo . Hence go must be the Lie algebra of all skew-Hermitian operators on Vo. This completes the proof.

3.1 Application. Let M be an irreducible symmetric Riemannian manifold. Let 0 EM be a point of M and let Vo be the tangent space at o. Furthermore let gil and Riikl be as usual, the metric and curvature tensors at o. We recall that if u = Xi and v = yi are two linearly independent vectors in Vo and [w] is the plane spanned by u and v, then the sectional curvature K[w] is defined by K[w]

=

see for example [1; 20]. More generally, let F2(VO) designate the space of 2nd order contravariant skew-symmetric tensors at o. Let P be the real projective space of line elements in F2(VO) and 11": F2(VO) - (0) ~ P the natural map of the non-zero elements of F2(VO) onto P. If a E F 2(V O), a :;c 0, we let [a] = 1I"a. Then the function K defined above can be regarded as being defined on the submanifold N of P corresponding to the set of 2-planes in Vo. Moreover, if we let w = xiyi yixi E F2(VO) ' then the notation above is consistent with this identification. Now K naturally extends to P where if a = (a ii ) E F 2 (VO), a :;c 0, then

103

A CHARACTERIZATION OF THE CLASSICAL GROUPS

119

We shall let KN be the restriction of K to N and we shall speak of it as the restricted sectional curvature. It is well known, of course, that KN completely determines K. Before proceeding it will be convenient to express the above in invariant notation. Let (u, v) designate the inner product in Vo as given by the metric tensor. Let a designate the Lie algebra of all skew-symmetric operators on V othat is, all operators A on Vo such that (Au, v) + (u, Av) = 0 for all u, v E Vo . In a we define the positive definite inner product Bo given by (A, B)o = - tr AB. Now let T: a ~ a be the operator on a defined by T(A) = B where

and where B = (b\) and A = (a';). Finally, let (T: F2(V o) ~ a be the isomorphism defined by lowering an index. That is, if a = a i ; E F 2 (VO), we let (T(a) = A where A = a\ = aikgk;' It is clear that if a ;e 0 and (T(a) = A, then (3.1.1)

K[a] =

.! (T A,

A)o. 2 (A, A)o

3.2. Now from the well-known identities ([1, (1.56)]) it is clear that T is a symmetric operator on a with respect to the given inner products. Hence T may be "diagonalized." That is, if ao is the kernel of T and X, , i = 1,2, ... , k are the non-zero distinct eigenvalues of T (necessarily real) and a, are the corresponding eigenspaces, we may write

where ri : a ~ a, , i = 0, 1, '" , n is the orthogonal projection of a on ai • It is clear, of course, that a = L:7~o a; is an orthogonal direct sum. Let go be the range of T. Of course go = L:7=1 a; . Now it is a simple consequence of the relation (3.1.1) that K as a function (analytic) on P has [a] as a critical point-that is, the differential, dK, vanishes at raJ-if and only if A is an eigenvector for T. We wish to show that for an irreducible symmetric space M, dim M > 1, K can never have a critical point in N with non-zero critical value unless K is constant (in which case, of course, every [a] is a critical point). Since it is well known that for an irreducible symmetric space, one must either have (a) K 2:: 0 or (b) K ::; 0 it follows that the maximal value of K in case (a) or the minimal value of K in case (b) can never be obtained in N unless K is constant. Interpreted in terms of the notation of §3.1 this means for such a space M an operator WE a of rank 2 cannot be an eigenvector of T with non-zero eigenvalue (that is, W f. a; , i = 1,2, ... ,n) unless T is a scalar multiple of the identity. Thus it suffices to show that if WE a; , i > 0, rank of W = 2, then T is a scalar multiple of the identity. 3.3. N ow if M is a Riemannian symmetric space then, as shown by E. Cartan (see e.g. [3; 265]) the curvature tensor satisfies the following algebraic identity:

104

120

BERTRAM KOSTANT

(3.3.1) By raising the index r we see easily that this is equivalent to the condition (3.3.2) for all A a'; (: go , the range of T. On the other hand, if ad A: II ~ II is the operator ad A (B) = [A, B], then it is straightforward to verify that (3.3.1) is equivalent to the condition that ad A commutes with T. Hence, in terms of the notation of §3.1, the algebraic identity (3.3.1) is equivalent to the condition ad A commutes with T for all A (: go . On the other hand, it is clear that an operator S on II commutes with T if and only if it leaves the eigenspace lli , i = 0, 1, ... , n, invariant. Hence if A (: go and B (: ll,. , then [A, B] (: lli. But then if A (: lli , B (: lli where i, j > 0, then [A, B] (: lli ( \ lli and hence [A, B] = 0 if i ~ j. We see then go is a subalgebra and that the lli C go are disjoint ideals in go . Conversely, of course, if go is a subalgebra of II and lli C go , i = 1, 2, ... , n, are ideals in go , then ad A commutes with T for all A (: go. It has to be observed, of course, that ad go leaves llc, (the orthogonal complement) invariant. THEOREM 3.3. The curvature tensor R.ikl satisfies the identity (3.3.1) if and only it the range go of T is a subalgebra of ll, and the eigenspaces Il. for i > 0 are ideals in go .

3.4. The fact that the range go of T is a subalgebra of II is, of course, well known for a symmetric space. In fact it was shown by Cartan that go is just the Lie algebra of the holonomy group, see [3; 223]. We obtain here the additional information that the eigenspaces of T, for the non-zero eigenvalues, are ideals in go . N ow since M is assumed to be irreducible symmetric, the holonomy algebra go acts irreducibly on Vo. We recall that we wish to show that if any ideal lli(i> 0) contains an operator of rank 2, then T is a scalar multiple of the identity. We now observe that it suffices to show that go = ll. Indeed if n ~ 4, then II is a simple Lie algebra and hence by Theorem 3.3 T can have at most one eigenvalue. If n = 4, then II = Ql + Q2 where Ql and Q2 are simple ideals both isomorphic to Lie algebra of SO(3). But neither Ql nor Q2 contains an element of rank 2. (In fact every non-zero element of Ql and Q2 has rank 4. This is true since exp Qi = Hi is isomorphic to the unit quaternions, and its action on Vo is equivalent to left multiplication on the quaternions in case i = 1 and right multiplication in case i = 2.) Thus if n = 4 and we show go = II under the assumption that T has an eigenvector in its range having rank 2, then Ql and Q2 are eigenspaces for the same eigenvalue. Hence even in this case T acts like a scalar. But now, assuming only that go contains an operator W of rank 2, it follows already from Corollary 2.7 that either (1) go = II or (2) Vo is even dimensional and admits a complex structure and a Hermitian metric with respect to which go

105

A CHARACTERIZATION OF THE CLASSICAL GROUPS

121

is the Lie algebra of all skew-Hermitian operators. We will now show that under the additional assumption that W E Q. for some i > 0, the second case (2) cannot occur unless dim Vo = 2 (in which case both (1) and (2) coincide). We begin with: LEMMA 3.3. Let c C go be the center of go. If c ~ 0, then for some i, c =

(t ••

Proof. Let h = go + Vo. We make h into a Lie algebra by (1) retaining the bracket relation in go , (2) defining [A, v] = - [v, A] = Av for A E go , V E Vo , and (3) for x = Xi and y = yi in Vo we have [x, y] = 2R i ikl X'yi Ego. From the standard identities on the curvature tensor and Theorem 3.3 it follows that h is a Lie algebra. In fact, as shown by Cartan, h is isomorphic to the Lie algebra of all infinitesimal motions on M; see e.g. [10; 80-83]. Consequently h is semisimple, see [5; 119]. Now if BI is the Killing form on h, that is, (X, Y)I = tr ad X ad Y for X, Y E h, it is immediate that go and Vo are orthogonal supplementary subspaces. Hence since BI is non-singular on h, it follows that its restrictions to go and Vo are, respectively, non-singular. But the restriction of BI to Vo is invariant under the action of go. Hence since go acts irreducibly on Vo , it follows that there exists a scalar b such that b(x, y)l = (x, y) for all x, y E Vo. Let B2 = 2bB l . When X = A Ego, we will write ad,A and A for the respective restrictions of ad A to go and Vo . We now show that for any A, B E go (3.3.3)

(A, TB)2

= (A, B)o .

For any x, y E Vo let W(x, y) E go be defined by W(x, y) = X'Yi - yiXi. Let l' = L~-I 1', so that 1': (t ~ go is the projection of (t on go. To show (3.3.3) it suffices by linearity to show (3.3.3) for all B of the form 1'(W(x, y)). But now T1' = T so that if B = 1'(W(x, y)), then TB = TW(x, y) = - [x, y] so that - (A, TB)2 = (A, [x, y])2 = (Ax, Y)2 = 2(Ax, y). But now 2(Ax, y) = 2a';x i y; = tr A· W(x, y) = - (A, W(x, y))o = - (A, 1'W(x, y))o = - (A, B)o . Thus - (A, TB)2 = - (A, B)o which proves (3.3.3). Now assume BE (ti ; then TB = XiB and hence by (3.3.3) Xi(A, B)2 = (A, B)o for all A Ego. Thus for all A Ego Xi ·2b tr ad A adB

= -tr

AB.

= -tr

AB

Conversely, if there exists X such that X·2b tr ad A ad B

for all A E go , then clearly B is an eigenvector of T with eigenvalue X. On the other hand, if B E c, then ad A ad B vanishes on go and hence tr ad A ad B = tr AB. Thus if X = -1/2b, we see that every element B E C is an eigenvector of T with eigenvalue -1/2b. Conversely, if BEg is an eigenvector of T with eigenvalue - 1/2b, then tr ad A ad B = tr AB for all A E go. Hence tr adgA adgB = 0 for all A E go. But this implies B E C since, admitting a faithful irre-

106

122

BERTRAM KOSTANT

ducible representation (its action on V o), go must be reductive. Thus, if c 'F- 0, there exists i such that - I/2b = Xi and Qi = c. This concludes the proof. Now it is a well-known result of E. Cartan that in case there exists a skewsymmetric J, such that J2 = - 1, and J commutes with go , then J I: go ; see [4; 250 and 257; IIIl. Hence if the case is that of (2) of Corollary 2.7, it follows that the J of (2), Corollary 2.7 belongs to go. In fact J I: C C g where c is the center of g. Now assume that (2) of Corollary 2.7 occurs. Since go acts irreducibly on Vo , it follows then that c is one-dimensional. Now by Lemma 3.3, c = Qi for some i > O. Let W I: Q; , j > 0, be of rank 2. Then if j = i, it follows that W = aJ for some number a. But then dim Vo = 2, in which case it is obvious that go = Q. Now assume that dim Vo > 2. Then j rE i and hence (W, J)o = O. We assert this implies W J = O. Indeed let U C Vo be the range of W. Then dim U = 2 and since J commutes with W, it follows that J leaves U invariant. But since J I: Q and since all skew-symmetric operators on a 2-dimensional space are proportional, it follows that there exists a scalar b such that bJ - W vanishes on U. Furthermore, since W vanishes on the orthogonal complement of U, this implies W(bJ - W) = o. But then tr W J = 0 implies - tr W 2= (W, W)o = 0 and hence W = O. But this is a contradiction. Hence only when dim Vo = 2, can (2) of Corollary 2.7 occur when we make the assumption that there exists W I: Q; for j > 0 of rank 2. But in this case we already have go = Q. We have proved ThEOREM 3.4. Let M be a Riemannian symmetric space. Let 0 I: M and let Vo be the tangent space at o. Let the ssctional curvature K be defined (at 0) as in §3.I on the projective space P associated with all bivectors at o. Then K takes a non-zero critical value on the submanifold N C P associated with the planar elements at 0 if and only if M has constant curvature.

REFERENCES 1. S. BOCHNER AND K. YANO, Curvature and Betti numbers, Annals of Mathematics Studies, no. 32, Princeton, 1953. 2. A. BOREL AND 1. DE SIEBENTHAL, Les S0U8-groupes ferm~s de rang maximum des groupes de Lie clos, Commentarii Mathematici Helvetici, vol. 23(1949), pp. 200-221. 3. E. CARTAN, Le~ons sur la Geometrie des Espaces de Riemann, Paris, 1946. 4. E. CARTAN, Sur une classe remarquable d'espaces de Riemann. I, Bulletin de la Societe MatMmatique de France, vol. 54(1926), pp. 214-264. 5. E. CARTAN, Sur une classe remarquable d'espaces de Riemann, I, II, Bulletin de la Societe MatMmatique de France, vol. 55(1927), pp. 114-134. 6. E. CARTAN, La. goometrie des groupes de transformations, Journal de MatMmatiques Pures et Appliquees, vol. 6(1927), pp. 1-119. 7. E. B. DYNKIN, The structure of semi-simple algebras, Uspehi Matematicheskih Nauk (N.S.)2, no. 4(20), (1947), pp. 59-127. American Mathematical Society Translation no. 17. 8. H. FREUDENTHAL, Zur Berechnung der Charakrere der halbeinfachen Lie8chen Gruppen. I,ll, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings of the Section of Sciences, no. 4, vol. 57(1954), pp. 369-376.

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A CHARACTERIZATION OF THE CLASSICAL GROUPS

123

9. F. GANTMACHER, Canonical representation of automorphism& of a complex semi-simple Lie group, Matematiceski Sbornik, vol. 47(1939), pp. 101-146. 10. B. KOSTANT, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Transactions of the American Mathematical Society, vol. 80(1955), pp. 528-542. 11. J-L. KOSZUL, Homologie et cohomologie des algebres de Lie, Bulletin de la Societe Mathematique de France, vol. 78(1950), pp. 65-127. 12. A. I. MALCEv, On semi-simple subgroups of Lie groups, Bulletin of the Academy of Sciences URSS, Series on Mathematics, vol. 8(1944), pp. 143-174. American Mathematical Society Translation no. 33. 13. Seminaire Sophus Lie, Ie annee 1954/1955, Theorie des algebre de Lie, Topologie des groupes de Lie, Ecole Normale Superieure, Paris, 1955. UNIVERSITY OF CALIFORNIA

108

Reprinted from the TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 93, No. 1, October, 1959 pp. 53-73

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT(1) BY

BERTRAM KOSTANT

1. Introduction. 1. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra of g. Let 7f).. be an irreducible representation of g, with highest weight A, on a finite dimensional vector space V". A well known theorem of E. Cartan asserts that the highest weight, A, of 7r" occurs with multiplicity one. It has been a question of long standing to determine, more generally, the multiplicity of an arbitrary weight of 7r". Weyl's formula (1.12) for the character of 7r" is an expression for the function X,,(x) =tr exp 7r,,(x) , xEf), on f) in terms of A and quantities independent of the representation. In the same spirit the author has always understood the multiplicity question to mean the following: Let I be the set of all integral linear forms on f). Let m" be the function in I which assigns to each integral linear form IIEI the multiplicity m,,(II) of its occurrence as a weight of 7r". Find a formula for the multiplicity function m" in terms of A and quantities independent of the representation. It is the purpose of this paper to give such a formula (1.1.5). Obviously a knowledge of the multiplicity function m" determines XA(x), xEf). That is, (1.1.1)

L

XA(x) =

mAC,,) exp ('" x).

vEl

On the other hand Weyl's formula asserts that

L (1.1. 2)

XA(X)

=

sg(cr) exp(cr(g

+ X), x)

crEW

--=-------

L

sg(cr) exp(cr(g), x)

crEW

where g is one half the sum of the positive roots and W is the Weyl group. Finding a formula for the multiplicity function m A in a sense then "accomplishes" the division indicated by the formula of Weyl. We hasten to addthis does not in any way detract from Weyl's formula since it still retains its overriding and quite remarkable feature of expressing what is in general a very complicated trigonometric polynomial on f) as a quotient of two relatively simple trigonometric polynomials. A direct interest in the multiplicity function arises from sources other than those mentioned above. Included are the following: Received by the editors April 26, 1958. (1) This research was supported by the United States Air Force through the Air Force

Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-79.

S3 B. Kostant, Collected Papers, DOI 10.1007/b94535_10, © Bertram Kostant 2009

109

54

BERTRAM KOSTANT

[October

(1) Let U be a compact connected Lie group and T';;. U a maximal toroidal subgroup. By virtue of the Frobenius reciprocity theorem the induced representation of U by a character of T is determined as soon as one knows mA(v) for all dominant X and a fixed suitable vEl. On the other hand such an induced representation is of special interest in algebraic geometry since as one knows it is equivalent to the representation of U defined by the natural action of U on the cross sections 9f the complex line bundle corresponding to v over the algebraic manifold (flag manifold) U/T. The knowledge of mA(v) for X~V supplements the Borel-Weil theorem with the information that 7rA occurs only on nonholomorphic cross-sections of the line bundle and does so with multiplicity mA(v). (See [1] for details.) (2) Concerning infinite dimensional representations a theorem of GelfandNeumark asserts that the restriction, to a maximal compact subgroup U of a complex semi-simple group G, of an irreducible unitary representation of G belonging to a nondegenerate series is given, as in (1), as soon as one knows mA(v) for all dominant X and a fixed suitable v. (See [5].) A means of computing mA(v) has been given by Freudenthal in [4]. The computation is based upon a recursive relation satisfied by the values mA(v), vEl, for a fixed X. This relation is an immediate consequence of what Freudenthal calls the Hauptformel. It is given as (see [3, 2.1 and 3.1])

(1.1.3)

L: mA(v +

kcp)·(v + kcp, cp) = mA(v) «A + g, A + g) - (v + g, v + g»

where the summation is over all positive integers k and all positive roots cp. For the purposes of finding a formula for mA(v), use of the relation (1.1.3) carries the repeated disadvantage of having always to divide by terms of the form «X+g, X+g) - (v+g, v+g» even in the case when v is not even a weight of 7r A• We could find no way in which (1.13) leads to a closed expression for mI.. Let P(p,) , p,EI, be the integer valued function on I defined by

pep,) = no. of ways p, may be partitioned into a sum of positive roots. I t follows from elementary considerations in representation theory that the inequality (1. 1. 4)

holds for all dominant X and all p,EI. Now one can show (and we exploit this fact) that, fixing p" for X sufficiently "far out" in the fundamental chamber and sufficiently far from the "walls" of the chamber the equality sign in (1.1.4) will always hold. It seems clear then that a formula for m A must necessarily involve the function P. It is the main result of this paper to establish the formula (1. 1. 5)

m,..(v) =

L: sg(u)P(u(g + A) aEW

110

- (g

+ v».

1959]

55

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

Putting A= 0 yields the following recursive relation for the partition function

P. (1.1. 6)

L

P(!J.) = -

sg(u)P(!J. - (g - ug»

aEW;D'~e

for ,u~O, ,uEI. (The recursive nature of (1.1.6) is further clarified when it is recalled that P vanishes outside the cone generated by the positive roots and g-ug lies in that cone. Also P(O) = 1.) 1.2. An auxiliary result is Theorem 5.1. Theorem 5.1 bears the same relation to the relation "totally subordinate" among representations, introduced in §4.4, as does a theorem of Dynkin (Theorem 4.3) to the relation "subordinate." Theorem 5.1 may be regarded as a weak generalization of the Clebsch-Gordan theorem. 2. Preliminaries. 1. Let g be a complex semi-simple Lie algebra of dimension n. Let 1 be the rank of g and let g be a Cartan sub algebra of g(dim g= l). Let B be the Cartan-Killing bilinear form on g. The value B assigns to vectors x, yEg will be denoted by (x, y). One knows that the restriction of B to g is nonsingular and hence one may identify g with its dual space. In particular ~, the set of roots of g with respect to g, is then a subset of g. Let go be the real subspace of g generated by~. Then one knows that go has real dimension l, the restriction Bo of B to go is positive definite, and

g = go + ig o is a real direct sum. Let W be the Weyl group of g regarded as operating on g. Elements x, yEg are said to conjugate under W if ux = y for some uE W. One knows that go is invariant under W. In fact it is only the action of Won go which is of interest to us. For each root cf>E~Cgo let g",~go designate the hyperplane orthogonal to cf> and therefore given by

g",

= {x

E go I (x, cjJ)

=

Also let R",E W designate the reflection of algebraically by R",x

for

go

O}. through

g",.

This

1S

given

2(cjJ, x)

= x - ---cjJ (cjJ, cjJ)

xEgo. The open set ill C 60 defined by ill

= go - U g", "'Ell.

is called the set of regular elements in go. The connected components D~ of ill are called open Weyl chambers. One knows that there are w of them where

111

56

BERTRAM KOSTANT

[October

w is the order of Wand that in fact they may be indexed by TV in such a way that if

is the decomposition of CR into its connected components D~ =u(DO) for any uEW and DO=D~, where e is the identity element of W. The closure D" of D~ will be called a closed \\leyl chamber or simply a Weyl chamber. Obviously one has

and D,,=u(D). Having fixed D--now called the fundamental chamber-among all the equally suitable Weyl chambers, we will say that an element xEgo is dominant if xED. We will say x is strongly dominant if xEDo. That is, x is strongly dominant if it is both dominant and regular. Each chamber D", uE W, decomposes A into a union of two disjoint subsets At and A;;- where A;;- = ( -1)A,t and c/JEAt if and only if (c/J, x) ~ 0 for all xED". Conversely, one knows that xED" if and only if (c/J, x) ~O for all c/JEA,t. Of course the inequality ~ becomes a strict inequality > when xED~. Write A+ and A- for At and A;-. The elements of A+ are called positive roots and they are in fact just the positive elements of A with respect to a suitable lexicographical ordering in go. We shall assume from now on that such an ordering is given in go. 2.2. Consider the lattice I (also, a discrete subgroup of go) of integral elements in go. By definition }LEI if and only if 2(}L, c/J)/(c/J, c/J) is an integer for all c/JEA. The set I is the set of all weights of all representations of g. Let n CA+, n = {aI, a2, . . . , al} , be the set of simple positive roots. The elements ai, i = 1, 2, ... , 1 form a basis of go. Let fi, j = 1, 2, ... , 1 be the dual basis to the basal elements 2a;/(ai' ai), i= 1, 2, ... , t. That is (2.2.1)

2(fj, ai)

- - - = fJ;j.

(a;, ai)

ThenjjEI,j=I,2, ·,1 and in fact these elements form a basis of I. That is, if }LEgo then upon writing I

j.J.

=

L

n;J;

}L E I if and only if the ni are integers. On the other hand 1; E D for j = 1, 2, . ,land in fact if xEgo then upon writing I

(2.2.2)

X

=

L i-I

112

c;fi

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

57

xED if and only if Ci~O, i= 1, 2, ... , land xEDo if and only if c.>O, = 1, 2, ... , t. Important in representation theory is the intersection ID=lrlD, the set of dominant integral elements in 1)0 and IDo=lrlDo, the set of strongly dominant integral elements in 1)0. 2.3. Now let 7r be a representation of 9 on the finite dimensional complex vector space V ... We shall always assume that the representation is complex linear. In such a case one knows that there is a unique decomposition of V.. as a direct sum of weight spaces V .. (JL), JLEI. That is

i

V .. =

L

V .. (/L)

pEl

where V .. (JL) is defined by V .. (IL)

= {v E V .. 1 7r(x)v

(IL, x)v for all x

=

E 1)}.

Of course V .. CJL),=O for only a finite number of JL. An element JLEI such that V.. (JL)'=O is called a weight of 7r. We will let A(7r) CI designate the set of weights of 7r. Now for any JLEI let m .. (JL) =dim V .. (JL). One always has m.. (/L) = m.. (u/L)

for any JLEI, uE W. Now in case 7r is irreducible the convex set in 1)0 generated by all the weights of 7r has as its extremal points a unique dominant weight X and all its conjugates {uX}, uE W. Anyone of these extremal points will be called an extremal weight. The weight X is the highest weight of 7r relative to any lexicographical ordering in 1)0 making A+ the set of positive elements in A. It is an already classical theorem, due to E. Cartan, that m .. (X) = 1 and that 7r is characterized by its highest weight. Furthermore, since any element vElD is the highest weight of some irreducible representation of g, we may use I D as the index set for the set of equivalence classes of all irreducible representations of g. In fact, for simplicity, for each XEID we choose a fixed irreducible representation of 9 with highest weight X and designate it by 7r).. The vector space for this representation will be designated by VA and, for simplicity, we will write V).(JL) for V ..,,(JL), m).(JL) for m..,,(JL) and A(X) for A(7r A). 3. The partition function P. 1. Now g admits the direct sum decomposition 9 = 1)

+L

(e
EA

where

eq"

(3.1.1)

cpEA, is a root vector associated with cpo Thus [x, e] =

(cp, x)e

for any xEfJ. Now let Beg) be the universal enveloping algebra (Birkhoff-

113

58

BERTRAM KOSTANT

[October

Witt algebra) of g. Let Zi, i=l, 2 , " ' , I be a basis of fJo. Then (see [7, Theoreme 1', p. 1-07]) the elements (3.1. 2)

(e1)~1(e2)~""

(eYT(z1)tl(Z2)t• ...

(ZI)tl(e-1)~1(e-2)~2

..•

(e-,)~T

form a basis of S(g). Here r is the number of positive roots. cf>i' i= 1,2, ... ,r, are the positive roots indexed so that cf>ii+1 and ~i' rh 'Y/k, 1 ~i~r, 1 ~j~l, 1 ~ k ~ r are non-negative integers, designating, of course, the powers of the corresponding basal elements. We shall need a simplified notation for this basal element. Towards this end let 0 designate the n-tuple (~i' rh 'Y/k) and write eO for the basal element (3.1.2). Let A designate the index set of all n-tuples 0 with non-negative integer coefficients. Thus the most general element PES(g) may be uniquely written p = 2: aoe o oeA

where a8 are complex numbers, only a finite number of which are distinct from zero. 3.2. Now since b..CI we can define a mapping of A into I as follows: the image of OEA is denoted by (0) and (0) is defined by (3.2.1)

(8) =

t

;=1

(~i -

71i)cf>i

where 0 is the integral n-tuple (~i' rh 'Y/k). The significance of (0) will be ap parent from the following: consider the infinite dimensional (purely algebraic) representation p of g on S(g) defined by p(x)p= [x, p] where xEg, PES(g). Then we observe that S(g) admits the direct sum decomposition (3.2.2)

S(g)

=

2: S~(g) ~eI

where

S~(g),

the "weight space for the weight

fJ,"

is defined by

S~(g) = {PE8(O)! [x,p] = (JL,x)P,xElJ}.

Indeed it is clear from (3.1.1) and (3.2.1) that e 8 ES(8)(g) and that in fact the set of all eO such that (0) = fJ, forms a basis of 8~(g). Since the set {eo}, OEb.., forms a basis of S(g) we have (3.2.2). Now any representation 7r of g on a vector space V". admits a unique extension to S(g) as a homomorphism of S(g) into the algebra of operators on V".. We shall always regard 7r as so extended. The significance of the decomposition as far as representation theory is concerned is that 7r(p) maps V .. (v)~ V .. (v+fJ,) for every PESI'(g). That is (3.2.3)

7f: S~(g) X V".(v) ~ V".(JL

+ v)

where 7r(p, v) =7r(p)v for PESI'(g), vE V,..(v). This is, of course, clear from the definition of SI'(g) and V".(v).

114

1959]

59

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

3.3. The space el'(g) is clearly infinite-dimensional. For many purposes it suffices to consider a particular finite dimensional subspace of el'(g) , the subspace generated by eq, where cf> is positive. Let n+, a maximal nilpotent subalgebra of g, be the Lie subalgebra spanned linearly by the root vectors eq, where cf>Et:.+. Then as usual one may regard e(n+) , the enveloping algebra of n+, as a subalgebra of e(g). Let A+ designate the subset of A consisting of all n-tuples ~i' 'f/k such that r;='l/k=O,j=l, 2,···, l, k=l, 2,···, r. We will use the letter ~ to designate elements of A+. Now we observe (applying [7, Theoreme 1', p. 1-07] again) that the elements ee, ~EA+ form a basis of e(n+). Let

r;.

e,,(n+) = e,,(g) II e(n+).

Then it is clear that if A+(M), MEl, is defined by A+(.u) = {~E A+ I (~> =

.u}

the elements eE, ~EA+(M) form a basis of e,,(n+). Furthermore it is also clear that e,,(n+) is finite dimensional. The function on I which assigns to each MEl the dimension of e,,(n+) plays a central role in this paper. Thus for any MEl let P(.u) = dim el'(n+) = number of elements in A+(.u).

We wish to make the following observations about P(M). First of all each may be regarded as a "way" of writing M as a sum of positive roots. Since repetitions of roots are permitted and the order in which the roots occur does not enter, ~ may be regarded as a "partition" of M into a sum of positive roots. Thus P(M) is, in effect, a partition function counting the number of partitions of M as a sum of positive roots. Upon writing ~EA+(M)

I

(3.3.1)

.u =

L:

niCl:i

i-1

it is obvious that P(M) =0, if for some i, ni is not a non-negative integer. Also note that P(O) = 1 (2). These facts are used in the recurrence formula given in §6 for P. We note more generally that if ni is a non-negative integer for all i then P(M) ~ 1. Indeed the formula (3.3.1) provides a way of writing M as a sum of simple positive roots. Since the simple positive roots are linearly independent there is only one such way of writing M. Now what was defined above for n+ we define similarly for n-, the Lie sub algebra generated linearly by all the root vectors e_q" cf>Et:.+. Let A- be (2) Recall that e(n+) contains the scalars and that ~i =li ='l/k =0 defines an element ~EA + such that eE= 1.

115

60

BERTRAM KOSTANT

[October

all n-tuples ~i' rio 'l/k such that ~i=rj=O, i= 1,2, ... ,r,j= 1,2,· ., t. We will use the letter '1/ to designate an element in A-. Analogously 81'(n-) has as basis the elements e~ where 'l/EA-(p,). Obviously dim 81'(n+) = dim 8_I'(n-) for any p,EI. 3.4. Returning to representation theory, for any }..EID let vAE VA be a weight vector belonging to the highest weight}... That is, (vA) = V A(}..). It is a well known and simple fact that every vector vE VA may be put in the form v=7r A(p)v A where PE8(n-). Indeed to prove this it suffices to know, (1), that the root vectors Cai and Lap where ai and aj run through the simple positive roots, generate 9 and, (2), that [e a " e"j]=5ijai. Furthermore it follows from (3.2.3) that for any p,EI every vector vE VA(}..-p,) may be written v=7r A(p)v A where PE8_I'(n-). That is, 7r).(8_I'(n-)v). = V).(X - JJ.).

It follows immediately then that (3.4.1) for any }..Eln . We will show that given any p,EI, }.. can be chosen so that the equality holds in (3.4.1). This and more will be needed in §6.2. 4. Theorems of Dynkin and Brauer. 1. For any}..Eln let}..*Eln be the highest weight of the contragredient representation to 7r A. 'vVe may always choose VA> so that VA> is the dual space to V). and 7r).o(x) is the negative transpose of 7r A(x) for any xEg. Now for any }..Eln we recall that A(}") = -A(}..*) and in fact mA(p,) =mA>( -p,). Note that this implies -}..* is the extremal weight of 7rx which lies in the chamber - D. It follows then that the one dimensional space V). ( -}..*) may be characterized by V).(-X*)

= {v E V A I7r).(x)v = 0 for all x E

n-l.

Now let 7r be any representation of 9 on a vector space V ... Define the subspace Z .. c V .. as follows:

Z". = {v E V .. I 7r(x)v = 0 for all x E n-} .

It follows immediately that (4.1.0)

dim Z .. = C(7r)

where C(7r) is the number of irreducible representations appearing in the decomposition of V .. into irreducible components. Now let }..l, }..2Eln and consider the case when 7r=7r).2®7rA~' the tensor product of the representations 7rA2 and 7rA~. It is well known that we may iden-

116

A FORMULA FOR THE MULTiPLiCITY OF A WEIGHT

1959]

61

tify the vector space V .. = V x• ® Vx~ with the space L( VXl' VA,) of all linear transformations A mapping V XI into V x• and that with respect to this identification 7r(x)(A)

= 7rx,(x)A - A7rA,(X),

But then we find that Z(7rx. ® 7rx~)

=

{A E L(Vx]! VA.) I 7rx.(x)A = A7rA,(X) for all x E n-}.

That is, Z(1I' X2®1I'X~) is the set of all intertwining operators for the pair of restriction representations 1I'Ali n- and 1I'x.1 n-(3). But then we note that if A EZ(1I'A2®1I'X~) 7rx.(p)A

(4.1.1)

=

A7rxI(p)

holds for all PE8(n-). On the other hand since every vector vE VAl may be put in the form V=1I'XI(P)VXl where PE8(n-) and (Vx) = VX(A), it follows that every A EZ(1I'x.®1I'x;) is uniquely determined by what it does to the single vector Vx,. Define the subspace WA.(A I ) C V x• by WX.(X 1) =

{v E

VA.

Iv =

Avx! for some A E Z(1I'X2 ® 7rx~)}.

Then, as we have just noted, the mapping (4.1.2) defined by (T(A)

= A1!Al

for A EZ(1I'A2®1I'A~) is an isomorphism onto. Recalling (4.1.0) we note in passing that we have proved LEMMA 4.1. Let AI, A2EID • For any finite dimensional representation 11' of g let C(1I') denote the number of irreducible representations occurring in the complete reduction of 11' into irreducible components. Then

(4.1.3)

C(7rA2 ® 7rA~) = dim W".(X,)

(4.1. 4)

~

dim V A2 .

We will be interested in the case when equality holds in (4.1.4). That is, when W".(AI) = V).. 2' Towards this end we wish to characterize the space WA.P'-l).

4.2. For any AEID and any vE VA let the left ideal S(v, A) in S(n-) be defined by 8(v, X)

= {p E S(n-) 17rx(P)V = o}.

(3) If rCg is a subalgebra and". is a representation of g denote by ". to f.

117

,..1 f

the restriction of

62

BERTRAM KOSTANT

[October

The following lemma is then an elementary fact in general ring theory: LEMMA 4.2. Let A!, A2EID ; then when vE VX2' VEWX2(Al) if and only if 8(vxl! AI) C8(v, A2)' That is, if and only if

7f)'t(P)VXl = 0 implies 1I"x.(p)v = 0

for any PE8(n-). Proof. If vE Wx.(Al) it is obvious from (4.1.1) that the condition of Lemma 4.2 is satisfied. Conversely if the condition is satisfied then setting A (1I).1(P)VXl)

= 1I).2(P)V

for all PE8(n-) defines (in a well defined way) an element A EZ(7rX2®7rX~) such that AVXl =V. 4.3. In [3], Dynkin introduces (Definition 3.2, p. 283) the notion of one representation being subordinate to another. In the case of irreducible representations, say 7rXl and 7r~2' Dynkin's definition is as follows: 7rx. is said to be subordinate to 7r~1 (or simply A2 is subordinate to AI) in case v~.E W~l ().2) (4). Recall (vx.) = V~2().2). Thus according to Lemma 4.2, A2 is subordinate to Al if and only if 8(v~J! Al)C8(v~2' A2)' That is, if and only if for all PE8(n-) (4.3.1) Dynkin then goes on to prove the following theorem (Theorem 4.3 below, Theorem 3.15 in [3, p. 285]), which asserts in effect that A2 is subordinate to Al if and only if (4.3.1) holds for a much smaller class of elements p. First however, we recall the following well known facts in representation theory. Let 7r be a representation of g on V ... Let cf>Etl, JLEtl(7r), v be any vector in V .. (JL) which is also an eigenvector of 7r(Lt/>et/». (It is known that one may find a basis in V .. (JL) which has this property.) Then (4.3.2)

2(cp, p.) (cp, cp)

=

p- q

where p is the smallest value of j such that 7r( (e_t/» i+ 1 )v = 0 and q is the smallest value of j such that 7r(~+l)V = O. For use later on we note the following easy consequence of (4.3.2). Let Mt/>(A) be defined by 2(cp, p.) Mt/>(X) = max - - ; I'E.1(X) (cp, cp)

then Mt/>(A) is the smallest value of j such that (4.3.3)

;+1 1I"x(et/> ) = O.

(') Note that in [3] extremal vector means weight vector for the highest weight, not any extremal weight.

118

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The equivalence of the two statements in the following theorem arises from the fact that if AEID and A= L:=l C;ji then by (2.2.1) and (4.3.2) Ci is the smallest value of j such that

i=1,2,···,l. THEOREM (DYNKIN) 4.3. Let AI, 'A 2 EI D. Then and only if'A I -A2EID. That is, writing

71""2

is subordinate to

71",,\

if

k = 1,2 7I"A2

is subordinate to

7I"AI

if and only if 1 2 C, ~ Ci

for i = 1, 2, ... , l. Equivalently andi=l, 2,···, 1

where

(VAk) =

7I"A 2 is

subordinate to

71",,\

if and only if for all j,

VAk(Ak), k=l, 2.

4.4. For any fJ-EI, 'AEID, vE VA let e_~(v,

X) =

e_~(n-)

n

e(v, X).

Then the inequality (3.4.1) in fact becomes (4.4.1)

Dynkin's theorem asserts that as the coefficients of A go up the right side of (4.4.1) goes down. We shall need the fact (proved later) that it can be made zero. Let 'AI, A2EID; we will now say that 'A 2 is totally subordinate to Al in case V A2 = WA 2 ('A l ). Obviously totally subordinate implies subordinate. From the point of view of general ring theory the notions of subordinate and totally subordinate are very easy to describe. Let k = 1, 2, then V"k may be regarded as a cyclic module over the ring e(n-), with cyclic vector VAk. Now associated with every cyclic module over a ring are two prominent subrings (1) the left ideal (here e(VAk' 'Ak)) of all ring elements which annihilate the cyclic vector and (2) the two sided ideal-now written as Jk-of all ring elements which annihilate the entire module. Obviously JkCe(VAk' 'A k). Now we observe that A2 is subordinate to 'AI when e(VAl' 'AI) s;e(VA2' A2) and totally subordinate to Al when e(VAI' 'AI) CJ2. Continuing from §4.1, obviously Lemma 4.1 implies

119

64

BERTRAM KOSTANT

[October

LEMMA 4.4. Let Xl, X2E I D; then X2 is totally subordinate to Xl if and only if

C(1I'X 2 ® 1I'X;) = dim Vx!. 4.5. If X2 is totally subordinate to Xl it follows immediately from Lemma 4.2 and (4.3.2) that 2(a" Al) 2(a;, fJ.) --->---(ai, ai) = (a;, ai)

for all }LEA(X2). Thus by §2.1, (4.3.2) and (4.3.3) we have LEMMA 4.5. Let XI, X2 EID. If X2 is totally subordinate to Xl then Xl - }LEID for all }LEA(X2) or equivalently 1I'A 1 (e!_aJ Vx,

= 0 implies 1I'x2(eL

aj )

= 0

for all j and all i = 1, 2, . . . , 1. 4.6. We will prove an analogue of Dynkin's theorem (Theorem 5.1) for the notion of totally subordinate (instead of subordinate). Theorem 5.1 asserts in effect that the condition of Lemma 4.5 is also a sufficient condition for totally subordinate. For this we need a theorem of Brauer. First, however, we wish to observe, LEMMA 4.6. Let XI, X2 EID then X2 is totally subordinate to Xl if and only if

X: is totally subordinate to Xi.

Proof. It is obvious that 1I'x: ®1I'XI is the contragredient representation to Hence C(1I'x; ®1I'x 1) = C(1I'X2®1I'A~). Since, of course, dim VA' = dim VA, the result follows from Lemma 4.4. 4.7. Let a(I) designate the group algebra over I. We admit into a(I) only functions h on I with finite support. It is also convenient to regard elements of a(I) as finite formal combinations of the elements of I. However, since the group operation in I is written additively we will designate the function (Dirac measure at v) which is 1 at v and zero at}L for all}L ~v by 5•. Thus when h is regarded as a formal combination of elements of I, h is written uniquely as

1I'A2®1I'X~.

h

=

:E a.B• • F.l

where only a finite number of the a. are distinct from zero. When regarded as a function, h(v) = a •. The function mA, defined in §2.3, which assigns to each vEl its multiplicity in 11'1\ is an element of a(I). Now for each vEl let F.E a (I) be defined by F. =

:E sg(uB).,•.

.,ew

Note that F.= sg(u)F.,. and as a function F.(}L) = sg(u)F.(u}L). Next we observe that F.~O if and only if vE
120

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65

deed if pE (R then (see §2.1) all the elements UP, uE Ware distinct and hence certainly F.~O. If, however, pEf)", for some cpEA then R",p=p, and since sg(R",) = -1, F.=FR ",.= -F. and hence F.=O. Thus it suffices to consider F. for regular P. In fact it suffices to consider only F., where wEIDo, that is, where w is strongly dominant. Indeed if pE(R there exists a unique wElD' such that F. = ± FOl. This is obvious since there exists a unique u such that upEIDo. Let w=uP; then F.=sg(u)FOl. An element hE a (I) will be called alternating in case h(~) = sg(u)h(u~)

for every p.EI. Let a.(l)ca(l) be the space of all alternating elements in a(l). The elements F. belong to a.(I) and are called elementary alternating sums. If hE a.(I) then since h(~) = -

1

L: sg(u)h(u~)

WaEW

where w=order (W) it follows that every element in a.(I) is spanned by elementary alternating sums. Hence from above every hE a.(I) may be uniquely written as h= L:OlEIDo bOlFOl where bOl=O except for a finite number of w. That is, the elements FOl' wEIDo form a basis of a.(I). Now let

Then it is a well known fact that gEl and in fact, 2(glai)/(ai, ai) = 1 for all i. That is

(4.7.1)

g=

h

+ f2 + ... + fl.

Thus from (2.2.2), gEIDo. In fact g is a very special element of IDo. It is the unique minimal element of IDo in the sense that w-gEID for every wEIDo. Conversely it is also clear that X+gEIDo for every XEI D • Thus if Tg designates the operation of translation by g in I then

(4.7.2) where the isomorphism is onto. This property characterizes g as an element of I. Thus we see that any hEa.(l) may be uniquely written as (4.7.3)

h

=

L:

AEID

aAF A+g

where aA = 0 except for a finite number of X. 4.8. Let Xl, X2 EID • In [2] R. Brauer has given a formula for determining the irreducible representations which occur in the tensor product of 'irA! and 'lrA 2 • This formula is given as Theorem 4.8 below. (In Theorem 4.8, however,

121

66

BERTRAM KOSTA NT

[October

we consider 7r},~®7r},1 instead of 7r},Z®7r},l. We do this because the formula for the former is more directly applicable for us.) Since 7r},;®7r},1 is symmetric in A: and Ai it is somewhat surprising at first to note that the formula is decidedly unsymmetric in Ai and A:. However, it is this feature which suggested the use to which this formula is put here. Recalling that m},o(p,) = m},( - p,), (see §4.1), we have THEOREM (BRAUER) 4.8. Let At, A2EID. Let n"A, AEID, be the multiplicity of the irreducible representation 7r}, occurring in the tensor product of 7r},; and 7r}'1 (of course n}, = 0 except for a finite number of A) so that 7r}.;

®

7r}.1

=

L:

n},7r}.

},EID

where the equality sign actually stands for equivalence. Now according to (4.7.3) let the coefficients a}" AEID, be defined by

(4.8.1)

5. A weak generalization of the Clebsch-Gordan theorem. 1. In the general case the formula (4.8.1) for the coefficients n}, has one serious drawback. That is because in general Al-p,EEID for p,EA(A2) and hence (see §4.7) there will be considerable cancellation in the contribution towards the coefficients on the right side of (4.8.1). But now if we assume A2 is totally subordinate to Ai then Lemma 4.5 asserts that this drawback is eliminated and hence in fact we can read off the coefficients n}, directly from the left side of (4.8.1). On the other hand we observe again that we need only assume Al-p,EID for all p,EA(A2) for this to be true. But under this weaker assumption we get the relation

L: n}, = L: }'eI

m},2(p,)·

I'EID

That is,

= dim

V>.;.

But by Lemma 4.6 this implies A2 is totally subordinate to Ai. We have thus proved the following theorem. The tensor product aspects of Theorem 5.1 may be regarded as a weak generalization of the Clebsch-Gordan theorem. A generalization, since if g is the Lie algebra of all 2 X 2 complex matrices of trace zero then for any pair At, A2EID, either Ai is totally subordinate to A2 or vice versa. Weak, since for general g not every pair Ai, A2EID are so related. Applying Lemmas 4.4 and 4.5 we have

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1959]

67

5.1. The following statements are all equivalent. Let AI, A2EID (1) A2 is totaUy subordinate to AI, (2) AI-MElD for all ME.!l(A2), (3) IfA1= L~=l c;f; then ci~Ma;(A2)' i=l, 2, ... ,l (see §4.3), (4) 'lrXt (tI-aJVXt = 0 implies 'lr X2(tI- a ,) = 0 for i = 1, 2, ... , l; j = 1, 2, (5) C('lrX2®'lrX~) =dim V X2 where C('lrX2®'lrX~) is the number of irreducible representations appearing in the decomposition of the tensor product 'lrA2 ®'lrA~ into irreducible representations. (6) For any 'hElD the representation 'irA appears m A2 (M) times, where M= Al - A, in the decomposition of the tensor product 'lrX~ ® 'lrAt into irreducible representations. THEOREM

5.2. Let g be as in §4.7. Consider now the "one-parameter" family of representations 'lrkg, k=O, 1, 2,···. But now for any AEID where A= L~=l cif. it follows immediately from (4.7.1) and Dynkin's theorem, Theorem 4.3, that 'irA is subordinate to 'lrkg if and only if k ~ maXi Ci. In particular we note that 'lrkg is subordinate to 'Ir(k+1)g for all k. Thus for all MEl (5.2.1)

Since as we recall (5.2.2)

it follows that the left side of (5.2.2) is monotone decreasing with increasing k. Next as an immediate consequence of (1) and (3) in Theorem 5.1 we have LEMMA

5.2. Let AEID. Then

k

'irA

~

is totally subordinate to 'lrkg if and only if

max Ma;(X). i

5.3. But now we can prove easily that the right side of (5.2.2) vanishes for k sufficiently high. From (5.2.1) it obviously suffices to show that for any PEBI'(n-) there exists k such that 'lrkg(P)VkuO=O. [This insures that we can always strictly drop the dimension of BI'(Vkg, kg) by choosing k large enough.] But now by Theorem 1 in [6] for any PEB(g) there exists AEID such that 'lrx(p) 0=0. In particular if PEBI'(n-) there exists vE VA such that 'lrx(p)vO=O. On the other hand by Lemma 5.2 for any k ~ maXi Ma;(A), 'irA is totally subordinate to 'lrkg. Thus from the definition of totally subordinate 'lrkg(P)VkuO=O. Thus we have proved LEMMA

5.3. Let MEl be arbitrary. Then there exists a positive integer N such

that for all

k~N.

Now for every O'E W let the subset T(O') C.!l+ be defined by

123

68

BERTRAM KOST ANT

[October

I

T(u) = {cp E Ll+ u- 1(cp) E Ll-l

and let s(u) EI be defined by

L

s(u) =

cp.

ET(v)

It is clear that if we write I

(5.3.1)

s(u)

L

=

bi(u)a;

i=l

then all the integers bi(u) are non-negative. If u¢e then b;(u) ~ 1 for at least one ~. Now observe that

L

u(g) = -1 ( 2

L

1/1 -

!l-E(A+-T(v»

cp) .

ET(v)

Hence we see that (5.3.2)

s(u)

= g - u(g)

for all uEW. In particular then it follows immediately from §2.1 and §4.7 that the elements s(u), uEW are all distinct. These elements of I will play a fundamental role in the remaining portions of this paper. In the definition of the function P on I we were concerned with the ways of expressing an element J.LEI in terms of the positive roots. Now we shall be concerned with expressing J.L in terms of the elements s(u), uEW. However, although in general there are far more elements s(u) than there are roots the matter is nevertheless simplified since now we will be concerned with the number of ordered ways of doing this-in fact ordered ways with a signature. Hence a recursion formula can immediately be given in this case. In the case of P(J.L), until we establish the equality given in Lemma 6.2 no recursive formula is apparent to the author. 5.4. For convenience write the elements s(u), uE W, as s;, i = 1, 2, ... ,w where w=order W. We may choose the ordering so that Sl=e and if s;=s(u) then -sg(u) = (_1)i. Now let r be the set of all finite sequences 'Y,

"Y

= (it,

i 2,

••• ,

iq)

such that i j is a positive integer and 2 ~ i j ~ w. For any J.LEI let the subset rl'cr be defined by

r I'

=

{"Y E r

IE

si;

=

It} .

Observe that since 'Y;~2 for all 'YEr it follows that rl' is a finite set. Now for any 'YEr define

124

1959)

A FORMULA FOR THE MUL TIPLlCITY OF A WEIGHT

69

Observe that if 'Y is the empty sequence, that is, q=O, then sg('Y) = 1. We can now define the function Q on I. For any p.EI let

Q(P) =

2: sg('Y)' 1'er"

If we write

we observe the following properties: (1) Q(J.L) =0 in case b; is not a nonnegative integer for some i, (2) Q(O) = 1, by a remark made above. Finally, (3), we note that if J.L¢O, Q satisfies the following recursive relation. (Here we revert to the original notation, s(u).) (5.4.1)

2:

Q(P) = -

sg(u)Q(P - s(u».

"'EW,cr~e

From (1) and (2) above, and from (5.3.1) we note that (5.4.1) provides an effective computation for Q. I t is especially interesting from the geometric point of view to write (5.4.1) in the form

2: sg(u)Q(J.L -

s(u» = 0

.. ew

for all J.LEI, p.¢0. 6. The multiplicity formula. 1. Let 'AEID • We now recall the formula of \Veyl for the character of the representation 1r~. For any xE1) Weyl's theorem (see [8] or [7, p. 19-07]) (6.1.1)

tr exp 'lrlo.(x) =

2: Fg+lo.(w) exp(w, x) 2: Fg(w) exp(w, x)

weI -",=------weT

where the (known) functions F.ECi.(I), pEl, are given in §4.7. On the other hand in terms of the (unknown) function mlo., obviously (6.1.2)

tr exp 'lrlo.(x)

=

2: m~(w)

exp(w, x).

weI

Setting (6.1.1) equal to (6.1.2) and clearing the denominator in (6.1.1) it follows that

Fa+"A

= Fg * mlo.

where * designates multiplication (convolution) in the group algebra G,(I). Thus recalling the definition of Fg and the usual expression for convolution

125

70

BERTRAM KOSTANT

(6.1.3)

L: sg(u)mA(w -

Fg+A(w) =

[October

u(g».

vEW

This immediate but none the less important consequence of Weyl's formula was first pointed out to us by Raoul Bott. Now let v=w-g and define G(v) = FII+A(w) = Fg+A(v

+ g).

Then GEa(J) has the following properties G(v) =

(6.1.4)

o for {sg(u)

v ;;tf. u(A

+ g)

for v = (T(A

- g,

+ g)

- g.

Substituting v+g for w in the right side of (6.1.3), solving for the case when u=e, and recalling that g-ug=s(u), we get (6.1. 5)

L:

mA(v) = G(v) -

sg«(T)mA(v

+ s(u».

G'EWiG'~e

Like (1.1.3) this sets up a recursive formula for mA(v). On the other hand (6.1.5) has one significant advantage over (1.1.3)-it does not necessitate the division of a term which might vanish. Now as in §5.4 write s(u) = Si where -sg«(T)=(-l)i and u;;tf.e implies i~2. Then applying (6.1.5) a second time we get w

mA(v) = G(v)

+ L:

w

(-l)ilG(v

+ Si) + L:

(_l)it+i2mA(V

+ Sil + Si2)'

In fact repeating the substitution k times we derive the relation k-l

111A(V) =

L:

w

L: w

+

L:

But now by (5.3.1) there exists an integer lvI such that for all k ~ lvI and any sequence 'Y= (iI, i 2 , • • • , i k ), W~ii~2, mA(v+Sil+Si2+ ... +Sik) =0. The same is true for G. Thus we can let k-t 00 and obtain from the definition of Q the relation (6.1.6)

mA(v) =

L: Q(w weI

Now recalling (6.1.4) we obtain

126

v)G(w).

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

1959]

71

6.1. Let XEI D , vEl then

LEMMA

m,,(v) =

:E sg(u)Q(u(g + X)

- (g

.. ew

+ v)) .

6.2. We will now proceed to prove the second major point of this paper, namely P=Q. LEMMA

6.2. The functions P and Q on I are identical.

Proof. Let p.EI be arbitrary. By Lemma 5.3 there exists an integer N such that (6.2.1) for all k~N. Now write p.

=

:E

biOli

i=1

and let N 1 =max i bi • Then by (5.3.1) for all integers

k~Nl

and all uEW,

UF-e

(6.2.2)

Q«k

+

1)(u(g) - g)

+ p.)

=

o.

This is clear since (k+1)(u(g)-g)+p.=p.-(k+1)s(u) and if we expand (k + 1)s(u) in terms of the (Xi, at least one of the coefficients must be negative. Let N 2 =max (Nit N). Apply Lemma 6.1 where X=kg, v=kg-p. and k~N2. Then

p. -

(6.2.3)

mko(kg - p.)

=

:E sg(u)Q(u«k + l)g)

- «k

+

1)g - p.)).

t7eW

But u«k

+ 1)g)

- «k

+ 1)g -

p.) = (k

+ l)(u(g)

- g)

+ p..

Hence by (6.2.2) all terms but one (u=e) drop out of (6.2.3). That is, mku(kg - p.) = Q(p.).

But then by (6.2.1) Q(p.) =P(p.). Q.E.D. The following is our main theorem. Summarizing from above we have proved THEOREM 6.2. Let g be a semi-simple Lie algebra with Cartan subalgebra g. Let leg be the discrete group of integral linear forms on g (see §2.2). Let P be the function on I,-the partition function- which assigns to every p.EI the number of ways p. can be partitioned into a sum of positive roots. (By a partition is meant multiplicities are permitted and the order is discounted. See §3.3.)

127

72

BERTRAM KOSTANT

[October

Let ~+ be the set of positive roots and let

g

1

L:

=-

~E
2

cpo

Let W be the Weyl group and let Seq)

=

g - q(g)

for any uE W. Then s(u) lies in the cone generated by the positive roots and for p,EI, p,~0, PCp,) satisfies the recursive relation

(6.2.4)

P(p.)

L:

= -

sg(q)P(p. - Seq»~.

crEW;tI"J'f;6

[In fact P can be defined by (6.2.4) when given that P(O) = 1 and pep,) = 0 for any p, having a negative coefficient when expanded in terms of the simple positive roots(5).] Let 7r be an irreducible representation of g. Write 7r = 7r>,. where "hElD (see §2.3) is the highest weight of 7r. Let vEl be arbitrary. Let m>,.(v) =0 in case v is not a weight of 7r>,. and in case v is a weight of 7r>,. let m>,.(v) be the multiplicity of the weight v. Then the number m>,.(v) is given by the formula

(6.2.5)

m>,.(v) =

L: sg(q)P(q(g + ><)

- (g

+ v».

uEW

REMARKS. In the author's opinion one of the most surprising aspects of the result above is the equality of P and Q. This equality, of course, enables us to establish formulas (6.2.4) and (6.2.5). [Note that when written in the symmetric form,

L: peP. -

s(q» = 0

uEW

for p,:;>"i0, the formula (6.2.4) becomes (6.2.5) for the case "h=0. As far as multiplicities are concerned the formula then asserts the obvious fact that for the identity representation the only occurring weight is the zero weight.] But now the proof of the equality of P and Q rests heavily on representation theory. Nevertheless the statement that P equals Q has nothing whatsoever to do with representation theory. In fact it is a statement in combinatorial analysis-asserting that the number of partitions of an arbitrary element p,EI where the parts are positive roots is equal to the number of ordered ways with signature plus one minus the number of ordered ways with signature minus one of writing p, with the elements g-u(g), uEW, u:;>"ie. We can find no direct proof to establish this equality (even in the case g=An). (6) This is clear since occurring among the s(.,.), selves. That is, s(Rai)=ai, i=l, 2"" ,l, see §2.1.

128

.,.E W, are the simple positive roots them-

1959]

A FORMULA FOR THE MULTIPLICITY OF A WEIGHT

73

BIBLIOGRAPHY

1. R. Bott, Homogeneous vector bundles, Ann. of Math. vol. 66 (1957) pp. 203-255. 2. R. Brauer, Sur la muUiplication des caracUristiques des groupes continus et semi-simples, C. R. Acad. Sci. Paris vol. 204 (1937) pp. 1784-1786. 3. E. B. Dynkin, Maximal subgroups of the classical groups, Amer. Math. Soc. Translations ser. 2 vol. 6 (1957) pp. 245-378. 4. H. Freudenthal, Zur Berechrung der Charaktere der halbeinfachen Lieschen Gruppen. I, Nederl. Akad. Wetensch. Indag. Math. vol. 57 (1954) pp. 369-376. 5. I. Gelfand and M. Neumark, Uber der Zusammenhang zwischen der DarsteUungen einer komplexen halbeinfachen Lieschen Gruppe und ihrer maximalen kompakten Untergruppe, Dokl. Akad. Nauk SSSR vol. 63 (1948) pp. 225-228. 6. Harish-Chandra, On representations of Lie algebras, Ann. of Math. vol. 50 (1949) pp. 900-915. 7. Seminaire "Sophus Lie,» Ecole Normale Superieure, Paris, 1955. 8. H. Weyl, Uber die Darstellungen der halbeinfachen Gruppen durch lineare Transformationen, Math. Z. vol. 24 (1926) pp. 328-395. UNIVERSITY OF CALIFORNIA, BERKELEY, CALIF.

129

Reprinted from the AMERICAN JOURNAL OF MATHEMATICS Vol. LXXXI, Number 4 October, 1959

THE PRINCIPAL THREE·DIMENSIONAL SUBGROUP AND THE BETTI NUMBERS OF A COMPLEX SIMPLE LIE GROUP.* 1 By BERTRAM KOSTANT.

1.

Introduction.

1. Let g be a complex simple Lie algebra and let G be the adjoint group of g. It is by now classical that the Poincare polynomial pG (t) of G factors into the form, !

(1.1.1)

Pa(t)

=

II (1 + t a,), ,=1

where l is the rank of g and the d. are odd integers. In this paper the integers m, (elsewhere, sometimes m, 1) defined by di = 2m. 1 will be called the exponents of g. No doubt one of the reasons the problem of finding the exponents turned out to be as difficult as it was, is that there was no way known by which these numbers could be determined from a direct examina· tion of the structure of g, particularly the root structure. The first procedure for extracting the exponents from the root structure of g was found by R. Bott. The proof of the validity of this procedure depends upon Morse theory. 2 A second and much simpler way, which we shall presently describe, of "reading off" the exponents from the root structure of g was discovered by Arnold Shapiro. (It is interesting that Shapiro discovered the procedure by misinterpreting the method of Bott.) However, even though one verifies that the numbers produced by this procedure agree with the exponents, unlike the case with Bott's method, the important question of proving that this " agreement" is more than just a coincidence remained open. The procedure is as follows: Let ~+ be the set of positive roots relative to some Weyl chamber in a Oarlan sub algebra of g. If cf> E 6,+ let a (cf» be the sum of the coefficients of cf> relative to the basis of simple positive roots. Let ble be the number of roots cf> such that a (cf» = le. Then ble - bk+1 is the number of times le occurs as an exponent of g. For example, if we apply this to the case where

+

+

* Received December 19, 1958. Research supported by Air Force Contract AF 49 (638) -79. • See R. Bott, "An application of the Morse theory to the topology of Lie groups," Bull. Boc. Math. France, t. 84 (1956), pp. 251-281. 1

973 B. Kostant, Collected Papers, DOI 10.1007/b94535_11, © Bertram Kostant 2009

130

974

BERTRAM KOSTANT.

g is Lie algebra of the spedal linear group SL (n, 0) then the fact that the number of matrix units e'j such that j - i = 7c, where 1 < 7c < n -1, is greater by one then the number of matrix units eij such that j - i = 7c 1 accounts for the fact that the exponents of g are 1, 2,' . " n-1. After Shapiro informed us of this counting device for the exponents we observed that it could be reformulated as follows: The principal three-dimensional subalgebra a o of g is a uniquely defined (up to conjugacy) three dimensional simple subalgebra (TDS) of g which can be readily distinguished from other TDS by its properties. [It was discovered almost simultaneously by Dynkin and de Siebenthal (see [6] and [13]) and later used extensively by these authors (see e. g. [7])]. Using standard facts in the representation theory of a TDS it is not difficult to show that the observation of Shapiro is equivalent to the observation that if we decompose the adjoint representation of a o on g into a direct sum of irreducible representations then the number of irreducible components is l and the dimensions of these components are ri., i= 1, 2,' . " l, where the d, are given by (1.1.1). However, this reformulation of the procedure still does not supply a proof. A second empirical procedure for finding the exponents was discovered by H. M. Coxeter. He recognized that the exponents can be obtained from a particular transformation y in the Weyl group, which he had been studying, and which we take the liberty of calling a Coxeter-Killing transformation, in the following manner (see [5]) : Let h be the order of y. Coxeter observed that (1) h satisfies hl = 2r, where r is the number of positive roots, (2) m, < h for all i and (3) the eigenvalues of yare wm" i= 1,2,' . " l, where w = e21fi /l&. A proof of (2) and (3) would provide, among other things, a proof of duality in the exponents mi observed by Chevalley (see [3], p. 24) since nonreal eigenvalues of y necessarily occur in conjugate pairs. Requiring (1) hl = 2r as the only empirically observed fact such a proof was recently obtained by A. J. Coleman (see [4]). A proof that hl=2r will be given in this paper. A second question posed in [4] of showing that h = 1 0 (tf!), where tf! is the highest root, will also be settled here. It will be the main result of this paper to establish a direct relationship between the principal TDS, its adjoint representation of g, and the transformation of Coxeter-Killing. The proof that the procedure of Shapiro yields the exponents is then a direct consequence of this relationship. A major role here is played by a particular conjugate class in G, the elements of which we call principal elements of G. It is shown that an element of G which induces a transformation of Coxeter-Killing on a Carlan subalgebra is necessarily a principal element. On the other hand a principal element of G

+

+

131

COMPLEX SIMPLE LIE GROUP.

975

belongs to the subgroup corresponding to a principal TDS and is sufficiently specialized in that subgroup so that the adjoint representation of a principal TDS on g is determined by its eigenvalues. That the conjugate class of principal elements is truly a distinguished one in G may be judged from the following characterization (one of several) of principal elements. Let A be an arbitrary regular element of G. Let k be the order of A (possibly O()). Then k ~ hand k = h if and only A is principal. An important role is also played here by two distinguished classes of g, the elments of which we have, respectively, called principal nilpotent and cyclic (the latter name is derived from the transformation properties of such elements in the case when g is the Lie algebra of SL (n, 0) ). As is the case with the principal elements of G these elements can be given simple characterizations (Corollary 5. 3, Theorem 9. 2 and Corollary 9. 3). In this paper §§ 1-4 are devoted to the theory of the general TDS. The main theorems here are Theorems 3.6 and 4.2. Both concern conjugacy questions. The first is an extension of a well known theorem of JacobsonMorosov. A corollary of it puts the conjugate classes of TDS in g in a canonical one-one correspondence with the conjugate classes of nilpotent elements in g. The second is implicit in the proof of a weaker theorem of Malcev. In § 5 the theory of the principal TDS is taken up. For the most part the theorems here are devoted to characterizing principal TDS among all TDS and principal nilpotent elements among all nilpotent elements. The result here which is used most often in the remainder of the paper is Corollary 5.3. The main results of the paper are given in §§ 6-9.

2.

Preliminaries and the complex three dimensional simple Lie algebra.

1. Let g be a complex semi-simple Lie algebra. Let nand l be respectively the dimension and rank of g. As usual the linear transformation y~

[x,y]

is designated by ad x and x ~ ad x is the adjoint representation of g on itself. If u is a Lie subalgebra of g the mapping x ~ ad x for x E u will be called the adjoint representation of u on g. We distinguish two types of elements in g. An element x Egis called nilpotent if ad x is nilpotent and is called semi-simple if adx is completely reducible, that is, (since we are dealing with

12

132

976

BERTRAM KOSTANT.

complex numbers) if ad x is diagonizable. 3 Only the zero element is both nilpotent and semi-simple. We recall that any element x contained in a Cartan sub algebra 1) C g is necessarily semi-simple and conversely every semi-simple element may be embedded in a Carlan subalgebra. (See e.g. [8], p. 119). For any element x E g let g'" designate the kernel of ad x. We recall that x is called regular when x is semi-simple and g'" is a Cartan subalgebra. 2.2. Let G designate the adjoint group of g. Since this is the only group associated with g that we shall consider the operation of exponentiation (Exp) will be always understood to go from g to G. Elements x and y in g are called conjugate if there exists A E G such that Ax=y. 2.3. The simplest complex semi-simple Lie algebra (up to isomorphism) is 0 1 , the Lie algebra of all complex 2 X 2 matrices of trace zero. In this case n = 3 and l = 1. One knows that any three dimensional complex semisimple Lie algebra is isomorphic to 01' By conjugating any element of 01 into Jordan canonical form the following is apparent: (a)

every non-zero element in

01

is either semi-simple or nilpotent,

(b) the set of all non-zero nilpotent elements in jugate class,

01

form a single con-

(c) if x, Y E 01 are semi-simple, x ~ 0, then y is conjugate to a unique, up to sign, scalar multiple of x. (Note also that the set of non-zero semisimple elements coincides with the set of regular elements in od. 2.4. Let 0 be a 3-dimensional complex simple Lie algebra. We recall further facts in the structure theory of o. Let x E 0 be a regular element. Then the eigenvalues of ad x are a, 0 - a for some non-zero complex number a. We may modify x by scalar multiplication so that a and - a take the values 1 and -1. This defines x uniquely up to conjugacy. We thus isolate a particular conjugate class in o. We thus isolate a particular conjugate class in o. The elements in this class will be called mono-semisimple. Let x Eo be a mono-semisimple element. Let

B+

Eo be a non-zero eigen-

• It is not difficult to show that this very same definition [of semi-simple and nilpotent elements] may be achieved using any faithful representation of g in place of the adjoint representation. This follows, e. g., from !.emma 5. 4 and well known facts in the representation theory of g.

133

977

COMPLEX SIMPLE LIE GROUP.

vector of ad x for the eigenvalue 1; e+ is unique up to a non-zero scalar. Then e_, an eigenvector of ad x for the value -1, is uniquely determined by condition [e+, e_J = x. One thus has the commutation relations

(1)

[x,e+J=e+

(2)

[x, e_ J = -

(3)

[e+, e_J = x

(2.4.1)

e_

for the basis x, e+, e_ of o. 2. 5. matrices

When

0 =

01

the elements x, e+, and e_ may be realized by the

__~(O0 1)0 '

e+- 2

e=2

_5(01 00) .

For each positive integer d there exists up to equivalence one and only one linear irreducible representation of 0 having that dimension. (For a complete treatment of the representation theory of 0 1 see [12, Expose no. 10].) To describe an irreducible representation 71'rl of g on a d-dimensional vector 1. It is of course space V first define the number k by the relation d = 2k to be noted that k is an integer if and only if d is odd. We may then find a basis vi> j ~ k, k -1,· .. , - k of V, where the vectors are each unique up to a scalar factor, satisfying the following condition:

+

The behavior of 71'rl(e+) and 71'rl(e_) on the one dimensional spaces (Vi) is given by

71'a(e+) (Vj)

=

(Vj+1)

71'a(e_) (Vi)

=

(vi-d

where Vk+1 = V-k-1 = o. Thus Vj is an eigenvector for 71'a(e+)7I'a(e_) and 71'a(e_)7I'a(e+), where in fact

and

It follows that with respect to the ordering Vk, Vk-1,· . ., V-k of the basal elements, when the latter are suitably modified by scalar multiplication, one obtains for 71'a(x), 71'a(e+) and 71'a(e_) the matrices

134

978

BERTRAM KOSTANT.

k

0

k-1 'lra(X)

=

o o

-7c

o

(1(27c))i o (2(2k-1»)i

o

o

o

(2k(1»i

o o

o (1(2k))1 0 (2(2k-1»)1

0

°

o

(27c(1))1

Several facts are to be noted.

0

Among those we shall require are

(a) The dimension d of V is odd or even according as the eigenvalues of 'Ira (x) are all integers or all half-integers. 4 (b) The eigenvalues of 'lra(x) all occur with multiplicity 1 and the real number j is an eigenvalue of 'Ira (x) if and only if d - (2 I j I 1) is a non-negative even integer.

+

+

( c) The number k, where 2k 1 = d, may be characterized as the highest eigenvalue of 'Ira (x). Furthermore, the one-dimensional eigenspace for this eigenvalue k may be characterized as the kernel of 'lra(e+). Now assume 'Ir is an arbitrary, not necessarily irreducible, representation of a on the finite-dimensional vector space V. One knows 'Ir may be decomposed into a direct sum of irreducible representations. It follows then that one knows 'Ir up to equivalence as soon as the dimensions of the irreducible components of 'Ir are given. That is, if nk denotes the number of such components having dimension 2k 1, then 'Ir is given when the sequence nk, k = 0, i, 1,' . . is known.

+

• We use the word half-integer to designate all numbers of the form m m is an integer.

135

+ t. where

979

COMPLEX SIMPLE LIE GROUP.

( d) The problem of finding the sequence nk can be reduced to an investigation of the kernel W C V of ?T (e+). In fact, it follows from (c) that dim W = no

+ ni + n + .. '. 1

Furthermore W is stable under ?T(x) and if wE W is any eigenvector of ?T(x), w may be embedded in an irreducible component of?T. Hence if leI, le 2 , ' • " lep are the eigenvalues of ?T(x) on W, the dimensions of the irreducible com1, 2le 2 1,' . " 2lep 1. (N ote that the ponents of ?T are respectively 2le l lei are non-negative.)

+

(e)

+

+

The space V admits a canonical direct sum decomposition

V=VE+ VO, where VE is spanned by eigenvectors of 7r(x) belonging to half-integral eigenvalues and VO is spanned by eigenvectors of ?T(x) belonging to integral eigenvalues. It follows immediately from (a) that VE and VO are both stable subspaces for the representation ?T and that in the complete reduction of 'IT IVE only irreducible representations of even dimension appear and the complete reduction of ?TI VO yields only irreducible representations of odd dimension. 5 (f) Now let V; be the eigenspace of ?T(x) for the eigenvalue j. Clearly dim V; = dim V _i' Furthermore, if j is non-negative, it follows from (b) that dim Vj=n;+ nj+1 n;+2

+

+ .. '.

The statement (f) has the following 2 consequences: (g) The dimension of V o, that is the dimension of the kernel of 7r(x) (nullity of ?T(x», equals no n 1 n 2

+ + +.

(h)

If j is non-negative,

dim V; - dim Vj+l = nj. Finally, we shall require (i)

If le is the maximal eigenvalue of 'IT(x),

2k

is a direct sum and

~ p=1

"I U

V p / 2 lies in the range of ?T(e+).

• If 7r is a representation on a vector space V and U C V is stable under'll", then denotes the representation on U obtained by restricting'll" to U.

136

980

BERTRAM KOSTANT.

3.

Nilpotent elements and TDS.

1. Let 9 be a complex semi-simple Lie algebra. Consider the question of determining all three dimensional simple subalgebras (TDS) in g. If a egis a TDS, then by considering the adjoint representation of a on 9 it follows from the representation theory of a outlined in § 2. 5 that any nilpotent element of a is necessarily nilpotent in 9 and any semi-simple element of a is a semi-simple element of g (Also see footnote 3). Since a has only semi-simple and nilpotent elements (see § 2. 3 (a) ), the question arises (1) which nilpotent and semi-simple elements of 9 can be embedded in a TDS and (2) how does one find all such sub algebras. We shall first consider the case of nilpotent elements. A theorem of Morosov asserts that every nilpotent element of 9 can be embedded in a TDS. (See [11].) However, his proof was incomplete. Later in [9] Jacobson gave a correct proof of this result. Since the proof leads into Theorem 3.6, we shall give it here. The proof requires Lemma 3.3. With the exception of the proof of Lemma 3. 3 the proof is the same as the one given by Jacobson.

3. 2. A famous result of Jacobson asserts that if A and B are linear transformations on a finite dimensional space V with the condition that [A, B] commutes with A, then [A, B] is nilpotent. If in addition A is assumed nilpotent, the following lemma (which is no doubt known) asserts that AB is nilpotent. 3.2. Let A and B be linear transformations on a finite dimensional space V. Assume A is nilpotent and LEMMA

[A, [A,B]] =0. Then AB is nilpotent. Proof. Let Vk be the kernel of Ak. We wish to show AB leaves Vk invariant. The result is obvious if k = o. Assume the result is known to be true for k = r. Let x E Vr+l ABx = [A, B]x

Apply Ar+l to both sides.

+ BAx.

Then

Ar+l(AB)x = [A, B]Ar+lx =

0

+ Ar+1BAx

+ k"(AB)Ax.

But Ax E Vr and by our assumption (AB)Ax E Vr. Hence ABx E Vr+1.

137

Thus Ar(AB)Ax =

o.

COMPLEX SIMPLE LIE GROUP.

Assume AB is not nilpotent. vector x E V, x:;l= 0 such that

981

Then there exists a scalar A, A:;I= 0 and a ABx=AX.

(3.2.1)

Let k be the smallest integer such that x E Vk+l Now AkABx=Ak[A,B]x AkBAx.

+

But AkB=BAk+k[A,B]Ak-l.

Thus

AkBAx = BAk+lX

=

0

+ k[A,B]AkX

+ k[A,B]AkX.

Hence AkABx= (k

+ 1) [A,B]AkX.

But by (3.21)

+

Hence [A, B]AkX = (Alk 1)AkX. Since A kX :;1= 0, this contradicts Jacobson's lemma asserting the nilpotence Q.E.D. of [A,B].

3.3.

Now for any x, y E g let (x, y)

=

tr ad x ad y

be the Cartan-Killing bilinear form B on g. Using the non-singularity of B on g we can now prove the following lemma. Lemma 3.3 is crucial in the proof that any nilpotent element e E g can be embedded in a TDS of g. It is clear that if e is contained in a TDS then e must lie in the range of (ad e) 2 since according to § 2. 3 (b), e can play the role of e+ in the commutation relations (2.4. 1) . In particular, it is interesting enough to observe then that for any non-zero nilpotent element e, (ad e) 2:;1= O. (The degree of nilpotency of ad e is greater than 2.) LEMMA

3. 3.

Let e E g be a nilpotent element.

Then e is in the range

of (ad e )2. Proof. The invariance of B under the adjoint representation implies that, for any z E g, ad z is skew-symmetric with respect to B and hence (ad z) 2 is symmetric. In particular, this is true for z = e. But now if A is a symmetric operator (with respect to B) on g and if Hi and KA are, respectively~ the range and kernel of A, then

138

982

BERTRAM KOSTANT.

By the non-singularity of B, then, to show an element z lies in RA , it suffices to show (z, y) = 0 for all y E K A' Letting.A = (ad e) 2, to prove the lemma it suffices to show that

[e, [e, y]]

(3.3.1)

0

=

implies (e,y) =0. But under the adjoint representation (3.3.1) becomes [ade, [ade,ady]] =0. Since ad e is nilpotent, we apply Lemma 3.2 to assert that ad e ad y is nilpotent. But then by definition of B it is clear that (e, y) = O. Q. E. D. 3.4.

By Lemma 3.3, e may be written as

(3.4.1)

[[f,e],e]=e

for some f E g. Let x = [f, e] so that [x, e] = e. LEMMA 3.4. Let gO be the kernel of ad e. Then gO is invariant under ad x. Furthermore if m is the smallest integer such that (ad e) m+l = 0, then m

II

(adx-p/2)

p=O

vanishes on ge. Proof. The space (ad e )Pg is the range of (ad e )p. We define a sequence of subspaces bp of gO, p = 0, 1,' . " m 1, where

+

(3.4.2) by letting First observe that ge is invariant under ad x. Indeed, if y E ge, [e, [x,y]]

=

[[e,x],y]

=-

[e,y]

=0. We now show that (3.4.3) Let y E bp.

Then y = (ad e)pz for some z E g.

Now since [x, 6] =

[ad x, (ade)p] =p(ade)p.

139

6

clearly

983

COMPLEX SIMPLE LIE GROUP. Thus [ad x, (ad e)p]z = py, or

[x, y] - p. y = (ad e)p[x, z] (3.4.4)

=

(ad e)p[[f, e]z]

=

(ade)P+1[z,f]

+ (ade)Padf[e,z].

But [(ad e)p, ad f]

1'-1

=

-l: (ad e) i ad x(ad e)l'-l-(

'=0 = ip(p -1) (ad e)l'-l_ pad x(ad e )1'-1. Applying this to [e, z] we obtain (ad e)p ad f[e, z]

=

adf(ad e)p[ e, z] + ip (p -1) (ad e )pz - p[x, (ad e)pz]

[f, [e,y]] +tp(p-1)y-p[x,y] = ip (p -1) Y - P [x, y].

=

But then (3.3.4) becomes

(p

+ 1) [x,y] -lp(p + l)y= (ad e)P+1[z, f]

or

[x, y] -tpy E Dp +1 • This proves (3.4. 3) . It then follows immediately from (3.4. 2) that m

II (adx-p/2) p=0

vanishes on ge. Q. E. D. It is an immediate consequence of Lemma 3.4 that ad x is completely reducible on ge and that its eigenvalues are restricted to non-negative integers and half-integers. In particular, what is essential for us at this point is COROLLARY 3.4. on ge.

The linear transformation ad x

+1

is non-singular

We can now prove THEOREM 3.4 (Jacobson-Morosov). Every nilpotent element of a complex semi-simple Lie algebra can be embedded in a TDS.

Proof. Let e be nilpotent, e oF 0, and let f and x be defined as in § 3. 4. 6 • The theorem holds when e = 0 once we know the existence of a single TDS. The existence of a TDS follows from the proof since g contains non·zero nilpotent elements.

140

984

BERTRAM KOSTANT.

In case [X, f] === - f we would be done, that is, x, e, f would satisfy the desired commutation relations. The problem is to modify f so that this relation is satisfied without destroying the relation (3.4. 1) . Even if [x, f] f =1= 0 we still have

+

[[x,f] +f,e] =0

+

as one easily checks. Thus [x, f] f E gO. But now by Corollary 3.4, since ad x 1 is non-singular on gO, there exists a unique g E go such that

+

[x,f] +f= [x,g] +g.

Then writing e+ for e and letting e_ = f - g it follows that [x, e+]

(3.4.5)

= e+

[x, e_] =-e_ [e+, e_]

=

x.

This proves Theorem 3. 4 as soon as one notes that x, e+ and e_ must be linearly independent. 3. 5. Any set of non-zero elements x, e+ and e_ in g satisfying the commutation relations (3.4.5) will henceforth be called an S-triple (to be written {x, e+, e_} ). The element x will be called the neutral element of the S-triple and e+ (resp. e_) will be called the nil-positive (resp. nil-negative) element of the S-triple. It is obvious that the elements of an S-triple form a basis of a TDS. Two S-triples are called conjugate if there exists A E G which carries one set onto the other. Given a non-zero nilpotent element e E g we wish now to find all S-triples which contain e as the nil-positive element. (By § 2. 3 (b) this yields all TDS which contain e). First we note the following corollary of the proof of Theorem 3.4. COROLLARY 3. 5. Let e E g be nilpotent, e =1= 0, then x and e are respectively the neutral and nil-positive elements of an S-triple if and only if (1) x is in the range of ad e and (2) [x, e] = e. Furthermore if x and e satisfy these conditions such an S-triple system is unique (and hence x and e are contained in just one TDS).

Proof. The first part of Corollary 3. 5 follows from the proof of Theorem 3.4 and the definition of x used in the proof (see (3.4.1». To prove the uniqueness of the nil-negative element assume that {x, e, fd and {x, e, f2} are two S-triples. It is obvious then that [e,fl-f2] =0

141

985

COMPLEX SIMPLE LIE GROUP.

so that

h-

f2 is an eigenvector of ad x

+ 1 on

ga.

By this implies f1 = f2.

Q.E.D. 3. 6. As a consequence of Corollary 3. 5 the problem of finding all S· triple systems containing e as the nil-positive element reduces to finding all elements x E g which satisfy the conditions of Corollary 3.5. Towards this end define, for e E g, the subspace ga=ade(g)

n ga,

the intersection of the range and kernel of ad e. We now observe that g. is a Lie subalgebra of g. Indeed, ge is a Lie sub algebra of g. Therefore it sufficies only to show that [u,v] E ade(g) if u,vE g•. Writing V= [e,w] we have, since u E go, [ u, v]

=

[u, [e, w]]

=

[e, [u, w]].

This proves [u, v] Ega. N ow let G. be the subgroup of G corresponding to the subalgebra g.. Note that among other things the elements of Ga leave e fixed. We can now state THEOREM 3.6. Let e E g, e oF 0, be nilpotent. Let g. and G. be as above. Then the elements of g. are all nilpotent (and hence the elements in G. are unipotent). Let x E g be such that x and e are, respectively, the neutral and nil-positive elements of an S-triple. Then the linear coset x g. of g. is the set of all neutral elements taken from all S-triples containing e as nil-positive element. Furthermore, any two elements in x g. are conjugate. Moreover the conjugation can be performed by an element in G. so that e is fixed under the conjugation. In fact for any A E G., Ax Ex g. and the map

+

+

+

defined by making A, A E G., correspond to Ax is one-one and onto. In other words (recalling Corollary 3.5) if {x,e,f} is the S-triple containing x and e, the map A~ {Ax,e,Af} sets up a one-one correspondence of the group G. onto the set of all S-triples containing e as nil-positive element. Furthermore Ax ranges over x g •.

+

142

986

BERTRAM: KOSTANT.

Proof. Assume {X, e, f} and {y, e, g} are S-triples and e nil-positive in both cases while x and yare neutral. Since [x,e] = [y,e] =e,

it follows that y - x E gO. But clearly [e, g - f] = y - x. That is, y - x E ad e (g) • Hence y - x E g. or y E x go. Oonversely, if y E x g., then clearly [y, e] = e and y E ad e (g). Thus applying Oorollary 3. 5 Y is a neutral element of an S-triple containing e. N ow according to Lemma 3.4 gO admits the direct decomposition

+

+

where m is the smallest integer such that (ad e) m+l = 0 and Op/2 is the eigenspace of ad x in gO belonging to the eigenvalue p/2. It is of course clear that (3.6.1) Let a be the TDS spanned by x, e and f. Now if7rd is an irreducible representation of a on a vector space of dimension d it is clear from § 2. 5 (i) that the eigenspace belonging to the highest eigenvalue (i(d-1», of 7ra(x) lies in the range of trd (e) if and only if d > 2. Now decompose g into irreducible subspaces under the adjoint representation of a on g and apply this fact to the irreducible components. Recalling 2. 5 (d) it becomes clear then that the subspace go of gO can be written m

(3.6.2)

gO=~OP/2 p=1

and hence, in particular, ad x is non-singular on go. One immediate observation from (3.6.1) and (3.6.2) is that go is a nilpotent Lie sub algebra of g. Furthermore, since the eigenvalues of ad x on go are strictly positive it is clear from a relation similar to (3. 6. 1) that ad w is nilpotent for every wE go.7 That is, the elements of ad go may be simultaneously triangulized with zeros appearing along the diagonal for every element. It follows then from well known facts concerning linear nilpotent Lie algebras that in such a case G. is closed, simply connected-that is, Ge is homeomorphic to Euclidean space-and the exponential map Exp: 7

g.~

Go

That is, if gj is the eigenspace of ad (J) on g for the eigenvalue j then clearly

[gj, lid C g,+j.

143

987

COMPLEX SIMPLE LIE GROUP.

is one-one and onto. That is, every A E G. may be uniquely written A for a unique w E g.. But then

(3.6.3)

Ax=x+ [w,x] +-Hw[w,x]]

=

Exp w

+ ....

Since g. is stable under ad x and since g. is a Lie subalgebra, all terms starting from the second on the right side of (3.6.3) lie in g. so that AxE x+ g•. Now let v E ge. Assume that there exists a unique element wj E g. such that (1) and (2)

m

Exp Wj(x) -

(x

+ v) Ep=j+1 ~ Op/2

Now let Zj+1 be the component of Exp wj (x) if

Wj+1 = Wj

(x

+ v)

in 0(j+1) /2. Then

+ (2jj + 1)zJ+1

it is clear that [Wj+1,x] = [Wj,x]-zJ+l. On the other hand, for i>l it follows from (3.6.1) that the components of (ad wj+d iX and (ad Wj) iX in OS/2 are the same for all s < j 1. Thus

+

i+1

Wj+1 E ~ 08 / 2 8=1

and

m

Exp Wj+1(X) -

(x

+ v) E. '=i+2 ~ 0

8 /2

and furthermore that in satisfying these conditions Wj+1 IS unique. If we define W1 = - 2v., where v. is the component of v in 0., then W1 uniquely satisfies (1) and (2) when j = 1. Thus we have proved inductively that there exists a unique W E g. such that

Expw(x) =x+v.

Q.E.D.

Note. It is useful to observe that the proof, above, of the statement that Ax ranges over x ge when A ranges over Ge depends essentially on just two facts, (1), the nilpotence of ge and (2), the non-singularity of ad x on ge. We can now supplement Theorem 3.4 with

+

COROLLARY 3. 6. Let e E g be nilpotent, e =1= 0, and assume e E 0 1 n 02, where 0 1 and 02 are two TDS. Then 01 and 02 are conjugate to each other. Furthermore, the conjugation can be chosen so as to leave e fixed.

144

BERTRAM KOSTANT.

988

P1·00f. According to § 2. 3 (b) we may find two S-triples each containing e as nil-positive element and which, respectively, are bases for a i and a2· Corollary 3. 6 then follows immediately from Theorem 3. 6. Q. E. D.

3. 7. Corollary 3. 6 is actually just a special case of the following corollary. The set of all TDS in g breaks up into conjugate classes under the action of G. Concerning these classes we have COROLLARY 3.7. The conjugate classes of TDS in g are in a natural one-one correspondence with the conjugate classes of non-zero nilpotent elements in g. The correspondence is established by associating to the conjugate class of a, a TDS in g, the conjugate class of any non-zero nilpotent element in a. That is, two TDS ai and a2 are conjugate if and only if ei and e2 are conjugates, where ei E ai, e2 E a2 and ei , e2 are non-zero nilpotent elements. Proof.

Follows immediately from § 2. 3 (b), Theorem 3.4 and Corollary

3.6.

Q.E.D.

4.

Semi-simple elements and TDS.

1. We now consider the semi-simple elements of a TDS and take up questions of conjugacy. Let {x, e+, e_} be an S-triple with x and e+, respectively, as the neutral and nil-positive elements. We wish first to determine all S-triples which contain x (necessarily as neutral element). Let a be the TDS spanned by x, e+ and L By considering the adjoint representation of a on g it follows from § 2. 5 that the eigenvalues of ad x on g are integers and half-integers. In fact, recalling § 2. 5 (f) and § 2. 5 (i), if gp/2 is the eigenspace of ad x for the eigenvalue p/2 and if k is the maximal value of ad x, then dim gp/2 = dim g-1l/2 and 2k

g= ~

gp/2.

p=-2k

In this case, however, we have the addtiional relation

(4. 1. 1) Since we are concerned with S-triples containing x, interest focuses on gi since any nil-positive element in an S-triple containing x obviously belongs to gi. In particular, e+ E gi. The question arises which other elements of Qi

145

COMPLEX SIMPLE LIE GROUP.

989

belong to S-triples containing :1: and whether such S-triples are conjugate to {:1:, e+, e_}. To settle this question we first consider go = g"'. It follows immediately from (4. 1. 1) that g'" is a Lie subalgebra. Let G'" be the subgroup of the adjoint group corresponding to the subalgebra gIll. It also follows from (4. 1. 1) that each of the subspaces gp/2 is stable under the adjoint representation of g'" on g and hence these spaces must be stable under Ga;. We are particularly interested in the action of G'" on gl'

4. 2.

N ow for each element e E gl it follows from (4. 1. 1) that ade: go~ gl'

Let T. be the restriction of ad e to go. Our interest now centers on what will be shown to be an important subset of gl' Define gl= {eE gl

I T. maps go

onto gd.

That is, e E l'it if and only if the rank of T. equals the dimensions of gl' We now observe LEMMA 4. 2A. A necessary condition that an element e E gl be the nil-positive element of an S-triple containing :1: is that e E gl' In particular,

e+E

(it.

Proof. Indeed, assume e and :1: belong to an S-triple. Let a' be the TDS which contains e and:1:. If we apply § 2. 5(i) to the adjoint representation of a on g it follows that gl is the range of ad e. On the other hand, it is clear from (4.1.1) that ad e (g) n gl = ad e (go) n gl' Hence T. must map go onto gl' Q. E. D. The following topological properties of III are needed for the proof of Theorem 4.2 (Here one is inspired by the use of regular elements in the usual proof of the conjugacy of any two Cartan subalgebras.). LEMMA

4. 2B.

The set gl is an open, dense and connected subset of gl'

Proof· It follows from Lemma 4. 2A that gl is not empty (e+ E gl)' Choose a basis of go and a basis of gl' For any e E gl set T.o equal to the dim go X dim gl matrix determined by T. and the given pair of bases. It is clear then that gl is the set of all e E gl such that at least one dim gl X dim gl minor of T.o is not zero. But it is an easily verified general fact that if Fh j = 1, 2,' . " m, are m non-zero polynomials on a complex vector space V then the complement -V to the set of common zeros in V of all the F J is open, dense and connected. Indeed, if u E V and v E V, then there are only a finite

146

BERTRAM KOSTANT.

990

number of complex scalars>.. such that F/(>..) =FJ(>..u+ (1->")v) vanishes for at least one j. Q. E. D. Now observe that gl is invariant under ()m. The major point in the proof of Theorem 3.4 is the observation contained in the following lemma. LEMMA 4.20. Let e E ih; then the orbit GllJe of e under the action of GIJJ is an open subset of gl (and hence of ?h).

Proof. The mapping A ~ Ae of GIJJ into gl is analytic. Thus it suffices to show that the differential of this mapping carries the tangent space to GIlJ at 1 onto the tangent space of gl at e. But the image of the former, under the differential, when translated to the origin of gl is just the subspace ad glJJ (e) of gl. But glJJ = go and since e E {h, this space coincides with gl, by definition of gl. Thus GlJJe is open in gt. Q. E. D. We have shown ()me is open in gl for any e E gl. But for el, e2 E gl, GlDel and GIJJ e2 are the same sets or else they are disjoint. But this fact taken together with Lemma 4.2B (the latter asserting the connectivity of fit) implies that there can be at most one orbit. That is, gl is itself a single orbit of GIJJ. But then recalling Lemma 4. 2A and observing that x is fixed under the action of GflI. we see that the following theorem has been proved. THEOREM 4.2. Let x E g be the neutral element of an S-triple {x, e+, e_}. (See § 3. 5.) As in § 2.1 let glJJ be the centralizer of x in g and let GIJJ be the subgroup of G corresponding to gllJ.

Define gl = {e E g I [x, e]

=

e}.

Then (4.2.1) for any e E gt. Let e E g. Then e and x are, respectively, the nil-positive and neutral elements of an S-triple if and only if (1) e E gl and (2) the map (4.2.1) is onto. Moreover, any two S-triples which contain x are conjugate to each other and the conjugation can be performed by an element in GID. In other words, if gl = {e E gl I (4. 2. 1) is an onto map}

then ?h is the conjugate class of e+ under the action of GID. That is

147

COMPLEX SIMPLE LIE GROUP.

991

Furthermore, ?h coincides with the set of all nil-positive elements taken from all S-triples having x as neutral element.

The statement of Theorem 4.2 gives a complete and simple description of the set of all nil-positive elements which "go" with a given neutral element. However, it is implicitly contained in Malcev's proof of the following corollary. Furthermore our proof of Theorem 4.2 although found without knowledge of Malcev's proof of Corollary 4. 2 (yet with the knowledge that it had been proved) amounts to only a technical simplification of Malcev's proof. Corollary 4. 2 provides the basis by which all the TDS in any complex simple Lie algebra have been classified (see [7], § 8). COROLLARY 4.2 (Malcev). Two TDS in 9 are conjugate if and only if any mono-semisimple element of one is conjugate io any mono-semisimple element of the other (see [10]). 4. 3. Now, continuing with the notation of § 4. 1 and § 4. 2, the subset ii1 in gl is of course only a part of the conjugate class of e+ in g. It does not seem likely that one can give a simple description of the entire class. Nevertheless, it is easy to describe a part of this class which is yet larger than gl (see Theorem 4.3, Theorem 4.3 is required for the proof of Theorem 5. 3) and we shall do this now. For any t = 0, 1,· .. , 2k let 2k

nt/2 =

~ gp/2·

p=t

It is clear from (4.1. 1) that nt/2 is a Lie sub algebra of g. Let Nt/2 be the subgroup of G corresponding to nt/2. We now have, in terms if the preceding notation,

THEOREM 4.3. The orbits Nox and Noe+ of e+ and x under the action of the group No are as follows: Nox=x+n. and

NOe+=gl+~.

In fact, G" is a subgroup of No and Nt is a normal subgroup of No. Moreover, the elements of N. are unipotent linear transformations of g. Furthermore, No can be written as a semi-direct product

13

148

992

BERTRAM KOSTANT.

(every element C E No is uniquely written C = AA', where A E Nt, A' E Gil) and the correspondence A~Ax

sets up a one-one mapping of Nt onto x A defines a mapping of No onto e+

+ nt while

the correspondence

~Ae+

+ tq.

+

Proof. Since no = g'" nt is a semi-direct sum (that is, [g"', nt] en.) it follows that every element C E No can be written, C = AA', where A E Nt and A' E Gill. On the other hand, if A E N. and y E gt/2, then Ay = y w,

+

21<

where w E

~ gp/2.

This implies A is unipotent. It also implies that Nt

n G'"

p=t+1

=

(l)and hence No is a semi-direct product of Nt and Gill. Now observe that

n. is stable under ad x and ad x is non-singular on nt. The proof that the correspondence A ~ Ax sets up a one-one mapping of Nt onto x + n. then proceeds in essentially the same way as the proof of Theorem 3. 6 (Recall that [nt/2' nt'/2] C n(t+t')/2 and see the note following Theorem 3.6) and we shall not repeat it. On the other hand, one cannot claim that the mapping of Nt into e+ tq defined by the correspondence A ~ Ae is one-one. This is because ad e+ annihilates non-zero elements in nt. But ad e+ maps gp/2 into g(p/2)+1 and applying 2.5 (i) to the adjoint representation of a on 9 it follows that

+

is an onto mapping. Proceeding then in a manner similar to the proof of Theorem 3.6 it follows easily that N. is mapped onto e+ n~ by the correspondence A ~ Ae+. Q. E. D.

+

As we have remarked, Corollary 4. 2 has provided the basis by which the conjugate classes of TDS have been classified (See [7], § 8). That is if x is a mono-semisimple element of a TDS, we know that x is a semi-simple element of 9 and that the eigenvalues of ad x are real (in fact are integers and half-integers). To classify the conjugate classes of TDS it suffices then by Corollary 4. 2 to find a fundamental domain for the action of G on the set of semi-simple x in g such that ad x has real eigenvalues and determine which in the domain are mono-semisimple elements of TDS. The Weyl chamber is such a domain and we shall presently consider it. (In view of Corollary 3.7 for the purposes of classifying conjugate classes of TDS it might be well to look for natural representative elements for the

149

COMPLEX SIMPLE LIE GROUP.

conjugate classes of nilpotent elements. gation has been undertaken).

5.

993

To my knowledge no such investi-

The principal TDS.

1. Let f) C g be a Cartan subalgebra which we shall assume is fixed once and for all. Let A be the set of roots with respect to f) and let e.p, cf> E A be representative root vectors so that we have usual direct sum decomposition g = f)

+~

(e.p).


Since B is non-isngular on f), we may assume A is embedded in f). That is, if cf> E A, then cf> may be identified with an element in f) by the relation

[x, e.p] for all x E f).

=

(x, cf> ) e.p.

One knows then if e.p and e_.p are normalized only in so far as

(5.1.0) as we shall assume, then (5.1.1) N ow let f)# be the real linear space in f) spanned by the roots. knows that

One

is a real direct sum. We recall that B is positive definite on f)# (see e.g. [12], p. 10-04). In particular then, (x, cf» is real for all x E f)# at all cf> E A. This means f)# can be characterized as the set of all elements x E f) such that ad x has real eigenvalues. Hereafter, if Y is any subset of f), we will let Y# = Y n f)#. Now for any cf> E A let

f)[cp] = {xE f) I (cf>,x) =O}. Then, clearly, if R is the set of all regular elements in f),

R# = f)- U f)[cf>].
The connected components of R# are called open Weyl chambers. Now assume a lexicographical ordering is given in f)#. Let A+ (resp. A-) be the set of positive roots (resp. negative roots). The ordering distinguishes a particular open Weyl chamber DO which can be defined by

DO = {x E f)# I (, x)

150

> 0 for

all cp E A+}.

994

BERTRAM KOSTANT.

The closure D of DO is called a Weyl chamber. It can be defined in the same way as DO except that one replaces the strict inequality > by the inequality > . The set D is a fundamental domain for the action of G on the set of all semi-simple elements y in g such that ad y has real eigenvalues. That is, if y is such an element there exists one and only one ([12], p. 16-08 and [7], Lemma 8. 2) element x in D which is conjugate to y. Now let IT = {1X1' 1X2, ' • " IX!} be the set of simple positive roots. The simple positive roots form a basis of f) and for every root q" upon writing

one knows that the coefficients 14 are integers which are all non-negative or all non-positive according as q,E A.+ or q,E A.-. We define the order o(q,) of q, by letting (5.1.2) Obviously, if q,l, q,2 and q,l

+ q,2 E A., then

(5.1. 3) Let

£i,

i

=

1, 2, .

. ,l, be the dual basis in h to the

IXj.

Since

(5.1.4) it is clear that

£i

E D.

More generally, if

(5.1. 5) then xED if and only if itt >0, i

=

1, 2,' . " l.

Indeed,

(5.1.6) Now assume xED is the neutral element of an S-triple {x, e+, e_}. Since the eigenvalues of ad x are integral multiples of i, it follows from that in the expansion (5. 1. 5) ai = imi for some non-negative integer mi' On the other hand, it is not possible for ai to be greater than 1. Indeed, since adx(e_) =-e_, it is clear that

where we emphasize the summation is over negative roots. Hence it follows from § 2. 5 (d) (with signs reversed. Also see Lemma 3.4) that if ai> 1, ad e_ (ea,) is a non-zero eigenvector of ad x with the positive eigenvalue itt -1.

151

995

COMPLEX SIMPLE LIE GROUP.

But then since xED, this implies [e_, eaJ is a sum of root vectors for positive roots. This, however, clearly contradicts the simplicity of ~i' We have proved then that ai = 0, i, or 1 for i = 1, 2,' . " 1. More than this we have (see [7], Theorem 8.3). LEMMA 5.1. Let the Weyl chamber D and the basis Ei, i = 1, 2,' . " 1 of q be given as above. Assume that xED is a mono-semisimple element of a TDS. Then

where for each i = 1, 2,' . " l, we have ai = 0, i, or 1. Furthermore, if X2,' . " Xb E D is the set (ordered) of all elements in D which happen to be mono-semisimple elements in at least one TDS (so that b < 3 1 ), then there are exactly b conjugate classes of TDS in g. In fact, if Xi is a monosemisimple element of the TDS a.; for i = 1, 2,' . " b, then the subalgebras a., i = 1, 2,' . ., b, are representatives of the conjugate classes.

Xl)

For each simple Lie algebra g Dynkin lists [7] which among the 3 1 - 1 choices are indeed mono-semisimple elements of a TDS. 5.2. We retain the notation of Lemma 5.1. Among the elements X;, j = 1, 2,' .. , b there is a distinguished one. This is the case when ai = 1, i = 1, 2,' .. , l. That is, for one of the X; all the a. (in (5. 15» take the maximal possible value. 5. 2. Let D and Xo E D be given by LEMMA

Ei,

i

=

1,2,' . ., l, be as defined in 5. 1.

Let

1

(5.2.1)

XO=~£i' i=l

Then Xo is a mono-semisimple element of a TDS Proof·

Since the elements of

~i

form a basis of

q#,

Xo may be uniquely

written I

Xo= ~ri~i' 1=1

N ow let Define (5.2.2)

C'h

i

=

1, 2,· . ·,1, be any 1 non-zero complex numbers. I

eo= ~ciea, .=1

and (5.2.3)

1

fo=~ (r./cj)e-a,. i=l

152

BERTRAM KOSTANT.

996

Clearly (ai, x o) = 1 for all i.

Thus

[xo, eoJ

=

eo

and On the other hand, since a. - aj is never a root, it follows from (5. 1. 1) that

[eo, foJ

=

Xo

Q.E.D. so that {xo, eo, fo} form as-triple. Let X o, eo and fo be as defined in the proof of Lemma 5.2. Let ao be the TDS spanned by the vectors. The TDS a o or any conjugate TDS is called (originally in [6]) a principal TDS of g. It and some of its properties were discovered by Dynkin and de Siebenthal (See [6J and [13]). We shall call the S-triple (or any of its conjugates) {xo, eo,fo} a principal S-triple. The matrices exhibited in § 2. 5 are an example of a principal S-triple in the Lie algebra of SL(d, C). One of the first properties we observe about the neutral element of a principal S-triple is that it is regular. In fact, Xo E f)# and for any cp E a clearly (5.2.4)

(see (5.1. 2) ). We shall call Xo or any element of g conjugate to Xo a principal regular element of g. Any element of the conjugate class of eo will be called a principal nilpotent element of g. One of the most significant ways in which a principal TDS is distinguished among all TDS is in regard to its adjoint representation on g. For any TDS a of g let n(a), nE(a) and nO(a) designate, respectively, the number of irreducible components occurring in the complete reduction of the adjoint representation of a on g, the number having even dimension and the number having odd dimension. THEOREM 5.2. Let a be any TDS of g.

a is principal if and only if n(a)

=

Then n(a) > l.

Furthermore

l.

Proof. Let {x, e, f} be an S-triple whose linear span is a. In fact, employing the notation of § 5. 1 we may assume xED C f). Since f) is contained in g'" (see §2.1), obviously dimg"'>l. But now according to 2.5(g) we have nO(a) = dim g"'. Thus

(5.2.5)

153

COMPLEX SIMPLE LIE GROUP.

997

which proves the first part of Theorem 5.2. N ow if a is principal x is regular and hence dim g'" = l. On the other hand, in this case all the eigenvalues of adx are integers (see (5.1.2)) so that nE(a) =0 (see §2.5(e)). Thus if a is principal, n(a) = nO(a) = l. Now, conversely, assume n(a) = l. But I

then by (5.2. 5) dim g'" = 1 which implies x is regular. Writing x =

~

ait:i as

(=1

in §5.1, it follows then that ai=i or 1 for all i. But since nO(a) =n(a), that is, n E (a) = 0, this means (see § 2. 5 (e)) the eigenvalues of ad x are all integral. Thus ai =1= t by (5.1. 6) and hence x

I

=

~ t:i

so that a is principal

.=1

by Corollary 4. 2. As a corollary of the proof we have

Q.E.D.

COROLLARY 5.2. Let a be any TDS of g; then nO(a) > l. Furthermore, n (a) ° = 1 if and only if the mono-semisimple elements of a are regular in g. Incase a is principal n E (a) = O. 5.3. The following corollary of Theorem 5.2 distinguishes the principal nilpotent elements in the set of all nilpotent elements in g. The proof follows immediately from § 2.5 (d) and Theorem 5.2. COROLLARY 5.3. Let e be a nilpotent element tn g; then dim ge > 1 and dim ge = 1 if and only if e is principal nilpotent. Now let {xo, eo,fo} be the principal a-triple defined as III § 5. 2. We will now apply the theory of § 4. 1 to this a-triple. First of all, since the eigenvalues of ad Xo are integral, we observe that gp/2 = 0 whenever p/2 is not an integer and for j a non-zero integer gj= ~ o(¢)=;

(5.3.1)

(e.p)

and We will write n for nt

=

n1 and

{l

for no.

n=

~

(e.p)

Then

¢Ell.+

(as is well known) is a maximal Lie subalgebra of nilpotent elements and f) n is a maximal solvable Lie sub algebra of g. Let H, N and be the subgroups of G corresponding to f), nand {l. Recalling that

{l =

+

a

eo

=

z ~ Cie""

0=1

154

998

BERTRAM KOSTANT.

where c. =F 0, i = 1, 2, . this case

. ,l it follows from Theorems 4. 2 and 4. 3 that in I

H eo = gl = {e E gl I e = }: aa,ea" where aa, =F 0, i = 1, 2,· .. , l}

.=1

and (since 11i = n2) (5.3.2)

Se O=gl+n2= {eE n I e=}: a4>e~, where a~=FO when ~€d+

cp = ai, i=1,2,· . ·,1}. We emphasize that (5.3.2) above implies that" almost all" the elements in the maximal subalgebra of nilpotent elements n are principal nilpotent (and hence lie in a single conjugate class). We wish to prove now that Se o contains every principal nilpotent in n, that is, we have the following simple characterization of those elements in n which are principal nilpotent. THEOREM 5. 3. elements given by

Let n C g be the maximal Lie stlbalgebra of nilpotent

Let e En,

be arbitrary. Then e is principal nilpotent if and only if i = 1, 2,· . ·,1.

a~

=F

°for cp

=

ai,

Proof. By (5. 3. 2) if e satisfies the condition stated in Theorem 5.3, then e is principal nilpotent. Conversely, let e E n be principal nilpotent and .assume aaJ = 0, for some j. Assume first that g is simple. Let 1/1 E .6. be the highest root. We recall I

two basic facts about the highest root.

One, upon writing 1/1 = }: qia, the

.=1

integers qi satisfy (5.3.3) and {5.3.4)

0(1/1)

> o(cp)

for any cp E .6., cp =F 1/1. Indeed, both of these facts are immediate consequences I

of the following single fact: If cp =

}: ti(Ji i=1

(5.3.5)

155

is any root, then

999

COMPLEX SIMPLE LIE GROUP.

for i = 1, 2, ... ,l. Finally, (5.3. 5) is a consequence of well known facts in representation theory; in fact, the adjoint representation of g is irreducible and by definition of 1/1, e", is the weight vector belonging to the highest weight, 1/1. One knows then that e.p, for any cp E ~, is obtained by applying polynomials (non-commutative) in the operators ad e-a" i = 1, 2,' . " l, to e",. This proves (5. 3. 5). Let I

q= ~ q.= 0(1/1). (=1

Now let Qo be as in § 5. 2 and consider the adjoint representation of Qo on g. It follows immediately from § 2. 5 (d) and (5. 3. 4) that (1) e", lies in an irreducible component 0 C g, (2) dim 0 = 2q + 1, (3) e_", E 0 and (4) any other irreducible has dimension less than dim 0. A direct consequence of these facts is that and where a =1= O. N ow since e is conjugate to eo it follows that (ad e) 2q =1= o. hand, since e E n, it follows that for any cp E ~ we can write

On the other

(5.3.6) and, moreover,

b~ =1= 0

implies

oa) > o(cp)

+ 2q.

But this together with (5.3.4) implies that (5.3.6) vanishes for all Thus (ad e) 2q ( e-tJt) =1= 0 and in fact, using (5. 3. 4) once more,

cp =1= - 1/1.

( ad e) 2qe-tJt = a' e", for some non-zero scalar a'. Now write e = e1 + e2, where e1 E gl and e2 E n2' Expanding (ade)2q= (ade 1 +ade 2)2q it is clear again from (5.3.4) that But for any i, writing ( ad e1) • ( e-tJt)

=

~ b'~e~,

C€a

it follows that since a,,} = 0, one can have b'~ =1= 0 only when upon writing

156

1000

BERTRAM KOSTANT.

we have tj = - qj. In particular, setting i = 2q this means qj = - qj or qj = O. This contradicts (5.3.3) and hence Theorem 5.3 is proved when g is simple. The general case follows immediately upon writing g as a direct sum of its simple ideals. The component of e in each ideal is necessarily principal nilpotent in that ideal by Corollary 5.3. Q. E. D. 5.4. A theorem proved in [1] (See [1], Remark p. 66.) asserts that every solvable sub algebra {l1 of g is conjugate to a subalgebra {l'l of {l. If in addition it is assumed that the elements of {ll are all nilpotent (hence {ll is a nilpotent Lie algebra by Engel's theorem), it is clear then that {l'l C n. In particuar, it follows then that every nilpotent element e Egis conjugate to some element e' E n. Actually, we don't need the result of [1] referred to above to prove this. For completeness, we observe that this follows from Lemma 5.1. LEMMA

5.4.

Any nilpotent element e Egis conjugate to an element

e' En. Proof. Applying Theorem 3.4 it suffices only to show that the nilpositive element of any S-triple containing Xj (using the notation of Lemma 5.1) as neutral element lies in n. But this is clear since Xj ED. That is, for cf>E fl., (xj,cf» =1 implies cf>E fl.+. As a corollary to Theorem 5. 3, its proof, and Lemma 5. 4 we obtain another characterization of principal nilpotent elements in case g is simple. COROLLARY 5.4. Assume g is simple. Let 1{1 be the highest 1'00t and let q = 0 (1{1 ) • Let e E g be any nilpotent element. Then e is principal nilpotent if and only if (ad e) 2q =1= O. However, if e is principal nilpotent (ad e)q+l = O.

5.5. Corollary 3.7 sets up a natural one-one relation between the conjugate classes of nilpotent elements and the conjugate classes of TDS. It is clear that in this correspondence the class of principal nilpotent elements corresponds to the class of principal TDS. Regarding the latter as distinctive among all conjugate classes of TDS will be given further justification then when it is shown that the former is distinctive among conjugate classes of nilpotent elements. The following corollary shows this very clearly. COROLLARY 5. 5. The set of principal nilpotent elements (a conjugate class of the adjoint group G) in g forms an open, dense and connected subset of the set of all nilpotent elements in g.

Proof.

Openness follows easily from Corollary 5. a by choosing a basis

157

COMPLEX SIMPLE LIE GROUP.

1001

of g and considering the (n - l ) X (n - l ) minors of the matrix defined by ad e, e nilpotent, with respect to the basis. Denseness follows from Theorem 5.3 and Lemma 5.4. Connectivity is immediate also since the set of principal nilpotent elements is an orbit of the group G. 5.6. Principal nilpotent elements behave like the "regular" elements in the set of all nilpotent elements. That is, one can make a strong case for the following analogy: The set of principal nilpotent elements is to the set of all nilpotent elements as the set of all regular elements is to the set of all semi-simple elements. We cite for example Corollary 5.3 and Corollary 5.5. In this analogy, between the semi-simple elements and the nilpotent elements, the Cartan subalgebra clearly corresponds to the maximum Lie subalgebra of nilpotent elements (see Lemma 5. 4 § 2. 1. Also all maximum Lie subalgebras of nilpotent elements are conjugate-see § 5.4). One knows that a regular element can be characterized by the property that it lies in one and only one Cartan subalgebra. Corollary 5. 6 asserts that also in this regard the analogy still holds. COROLLARY 5. 6. Let e E g be nilpotent. (One knows that e can be embedded in at least one maximal Lie subalgebra of nilpotent elements.) Then e is principal nilpotent if and only if e lies in one and only one maximal Lie subalgebra of nilpotent elements of g.

Proof. By Lemma 5. 4 we may assume e E n (using the notation § 5.4) . Assume that e is not principal. We will prove e is contained in a second (different) maximal Lie sub algebra of nilpotent elements n/. N ow we may write

By Theorem 5. 3 since e is not principal, aa, now it is known (and easy to verify) that

=

0 for some value of i.

But

is a maximal Lie subalgebra of nilpotent elements of g. (In fact, n is carried onto n' by any element of G which (1) leaves ~ invariant and (2) whose restriction to ~ is the reflection Ra,-see § 7.1). Obviously e E n/. N ow assume e is principal. In fact, we may take e = eo, where we use the notation of § 5. 2. Assume eo E n/, where n' = An, A E G. We shall use

158

1002

BERTRAM KOSTANT.

the prime (') on previous notation to indicate the effect of conjugation by A. Applying Theorem 5.3 to n' it follows that

eo =

~

a.pe.p',


where a.p ¥= 0 for cf> = rt.;, i = 1,2,· .. , I. A1 E N' such that

N ow by Theorem 4.3 there exists

I

A1eo = ~

aai ea /.

i=l

But then since xo' (= Ax ) and A 1 eO are clearly the neutral and nil-postive elements of an S-triple (see proof of Lemma 5.2), it follows from Theorem 3.6 that However, since x o' is regular upon applying § 2.5 (d) and (5.1. 2) in the case of this S-triple, it follows that gA l 6 0 en'. But then by Theorem 4.3 there exists A2 EN' such that A2AIXO = x'o. But now since Xo is regular, we must have A 2 A 1 l) = l)'. But this of course implies l) C s' because A 1 -IA 2-1 E S'. But this means s' and hence n' = [s', s'] are stable under ad l). But then n' must contain and in fact must be spanned by root vectors associated with l) (since obviously n'

n l) =

I

0) . But eo =

~.

c;e a , E n' and

Ci

¥= O. Thus eat En',

i=l

i = 1, 2,· .. , I. Hence since the eal generate n, it follows that n en'. But then of course n = n'. Q. E. D. 5. 7. We continue with previous notation. Now consider goo, the kernel of ad eo. By Corollary 5. 3 dim g60= I.

+

+

In the special case when g is the set of all (l 1) X (I 1) complex matrices of trace zero one sees easily that g60 is a commutative Lie algebra (of nilpotent matrices). This and other evidence suggested to us that perhaps g60 is commutative in the general case. However, since among other things we were unable to construct a "useable" basis of g60 we could not settle the question using purely algebraic methods. Nevertheless, it is true that g60 is commutative (Corollary 5.8) in the general case. The proof which we have found is very simple but uses limit arguments. It was Corollary 5.8 which orignally suggested the validity of Theorem 6.7. Theorem 5.7 or Corollary 5.7 may be regarded as a generalization of the fact that a Cartan sub algebra of a semi-simple Lie algebra is commutative.

159

COMPLEX SIMPLE LIE GROUP.

1003

THEOREM 5. 7. Let 9 be a complex semi-simple Lie algebra of rank l. Let y E 9 be arbitrary. Then g1l contains an 1 dimensional commutative Lie subalgebra.

Proof. It is well known that the set of regular elements in 9 is dense III g. Thus we may find a sequence y'n, n = 1, 2,' .. of regular elements converging to y. Now consider the Grassmann manifold of alll planes in g. This, of course, is compact and hence we may find a subsequence Yn of the sequence y'" with the property Yn ~ y and the Cartan subalgebras g1l.. converge to an l-plane u in the Grassmann manifold. Now if Wi, i = 1, 2,' . " l, is any basis of u we may find elements Wi" E g, n = 1, 2,' . " i = 1, 2,' . " l, such that Win E g1lft and Win ~ Wi as n ~ 00 for i = 1, 2,' . ',1. Since [y", Wi"] = 0, it follows immediately by taking the limit that u C g1l. But [w.;n, wt] = O. Again taking the limit this obviously implies u is commutative. Q. E. D. COROLLARY 5. 7. tative. 5. 8.

Let y E g.

Assume dim g1l = 1.

Then g1l is commu-

Corollary 5. 7 and Corollary 5.3 imply

COROLLARY 5. 8.

Let e be a principal nilpotent element in g.

Then ge

is commutative. 6.

The principal element of G and the duality theorem.

1. We shall assume from now on that 9 is simple. The theorems to be proved are either true in the general semi-simple case or can be obviously modified to be true in that case. The extension from the simple case to the semi-simple case in any event is immediate. We consider only the former mainly for notational simplicity. Recall what is meant by a compact form f of g. This may be defined as a real Lie sub algebra of 9 with the property

g=f+if

(1) is a real direct sum. (2)

(That is, f is just a "real form" of 9 and

The restriction of B to f is negative definite. One immediate consequence of (2) is

(a) every element of f is semi-simple. Next we recall some facts in the Cartan subalgebra theory of f (which is somewhat different from that of g).

160

1004

BERTRAM KOSTAN'f.

+

(b) If t is a Cartan sub algebra of f, then f.lt = t it is a Cartan subalgebra of g and f.lt # = it. (c) A Lie sub algebra t of f is a Cartan subalgebra if and only if it is maximal commutative. An immediate consequence of (c) which we shall soon use is (d) Any commutative Lie subalgebra u of f can be imbedded in a Cartan sub algebra of f. Corresponding to the adjoint operation (conjugate transpose) in the space of matrices we introduce a *-operation in g by defining for any z E g

z*=x-iy, where z is written for x, y E if. Generalizing the notion of a normal matrix or normal operator we will say z Egis normal with respect to f whenever

[z,z*] =0

+

Writing z = x iy, x, y E if it is clear that z is normal if and only if [x,y] =0. It is well known of course that a normal matrix is diagonizable (semisimple). The following generalization of this fact (and also of (a)) is a useful criterion for semi-simplicity in g.

Let z, an element of g, be normal with respect to a compact form f of g. Then z is a semi-simple element of g. LEMMA

6.1.

+

Proof. We may write z = x iy, where ix, iy E f. But now since z is normal [ix, iy] = o. Thus by (d) there exists a Cartan subalgebra t of f which contains ix and iy. But by (b) 91 = t it is a Cartan sub algebra of g. But then z E f.lt and hence z is semi-simple. Q. E. D. When the root vectors, relative to a Cartan sub algebra, are suitably normalized (Weyl's normal form) Weyl has given a basis of a compact form of g in terms of these root vectors and the Cartan subalgebra. In previous sections we required no other normalization of the eq>, cp E~, other than (5. 1. 0). We will assume from here on, unless specified otherwise (in Theorem 8.4), that the root vectors are normalized into Weyl's normal form, with the exception that (eq>, e_, e_q> ) = -1. (In such a normal form one may choose the root vectors ea"

+

161

COMPLEX SIMPLE LIE GROUP.

1005

i = 1, 2,' . " l, arbitrarily and hence we need not regard

eo as having been altered). Then one knows that the linear span, with real coefficients, of iU# and the vectors et/> - e_t/>, iet/> ie_t/> for all cp Ea+ is a compact form f of g. (See remark, p. 11-11 in [12J). One sees easily then that (et/» * = e_t/> or

+

(6.1.1) for any set of complex numbers at/>, cp E a. 6. 2. We recall now (see § 6. 1) that g is simple. Let if! E a be the 1 roots obtained by adjoining highest root.s Let n Q C a be the subset of l - tf; to the simple positive roots. That is, n Q = n u (- if!). The notion of simple root and highest root are notions which of course are relative to the choice of a lexicographical ordering in U# or rather to the choice of a Weyl chamber (see § 5.1). We will say then that the roots in n Q are Q-simple relative to the chamber D. (We shall not require the fact but it can be shown that the number of Weyl chambers which give rise to the same set of Q-simple roots is equal to order of the fundamental group of G; that is, to the order of the center of the simply connected covering group of G.) Now an element z E g will be called cyclic if there exists a Cartan subalgebra Ul and a set n 1 Q Cal of Q-simple roots relative to some Weyl chamber Dl in Ul # such that z can be written

+

(6.2.1) where the at, ~ E n 1 Q, are non-zero complex numbers and the e~ are root vectors for fh corresponding to the roots ~ E a 1 • 9 Cyclic elements playa major role in the remainder of this paper. If z is given by (6. 2. 1), observe that in effect we have formed the cyclic element z by adding a-'/Jle-'/Jl to a principal nilpotent element, where tf;l is the highest root in a 1 relative to D 1 • It shall be shown that not only does this destroy nil potency but the cyclic elements are in fact regular. First we shall need LEMMA

6.2.

Let the cyclic elements z, z' E g be given by

L

Z=

afJefJ

fJ(ITO

and z' =

~ a'fJefJ,

fJ (ITO

8 The properties of t/I which will be required are all consequences of (5.3.5). • The set III is the set of roots associated with the Cartan subalgebra lit.

162

1006

BERTRAM KOSTANT.

where the coefficients ap, a' p are all arbitrary non-zero complex numbers. Then if H is the subgroup of G corresponding to ~ there exists an element A E H and a non-zero scalar A such that AZ=Az'. Proof. Let

£.;,

i

=

1, 2,· . ., l, be as in § 5. 1. Let y E ~ be given by

,

y = ~ Log (a'a.!aa,)£.;. i=1

Clearly Exp y E Hand ,

z

Expy( ~aa,ea,) =~a'a,ea,. ":::::1

-£=1

Now let c be any complex number and let Xo be given by (5. 2. 1). Then clearly ,

Exp(y

(6.2.1) Now let q =

+ cXo) ( ~ aa,ea,) = i=l

0 ("')

z

eC (

~ a' a,ea.). -£=1

as in § 5. 3. But then Exp cXo ( e_f/1)

=

e-cqe-,/!

and if b is defined by Exp y ( e-'/!) = be-,/!, then b =1= 0 and (6.2.2) N ow choose c so that Then (6.2.3) Hence if A = eO, A = Exp(y and (6. 2. 3) imply

+ cXo), one has A E Hand (6.2.1), (6.2.2), Q.E.D.

An immediate consequence of Lemma 6. 2 is the following theorem which asserts that up to scalar multiplication the set of cyclic elements forms a single conjugate class in g. 6. 2. Let z and z' be any two cyclic elements in g. Then there exists a scalar A, A =1= 0, such that z and Az' are conjugate to each other. THEOREM

Proof. If D l is a Weyl chamber in ~l #, where ~l is a Cartan subalgebra, there exists A E G such that AIh = Ih and ADl = D. This fact together with Lemma 6.2 proves Theorem 6.3.

Q. E. D.

163

1007

COMPLEX SIMPLE LIE GROUP.

6.3.

Lemma 6.1 is needed solely to prove the following lemma.

LEMMA

6.3. Cyclic elements are semi-simple.

Proof. N ow if we write z

if; = ~ qi~i,

(6.3.1)

;'=1

recall that the coefficients qi are positive integers. is defined by

(See (5.3. 3) ). Thus if e1

z

~

e1=

(q.)l;ea,

.=1

and Zl is defined by then e1 and Zl are, respectively, principal nilpotent and cyclic elements. To prove Lemma 6.3 it suffices by Theorem 6.2 to prove only that Zl is semi-simple. But to prove that Zl is semi-simple, by Lemma 6.1 it suffices only to prove that Zl is normal with respect to f. N ow by (6. 1. 1) and z

~

e1* =

(q;,)l;e--«,.

;'=1

But, obviously then, since if; is the highest root

[e 1,e.p]

=

[e1*,e_.p] =0.

Thus But z

[e1, e1*]

=

~

z

q.[eap e_a,]

=

;'=1

by (5. 1. 1) and (6. 3. 1).

;'=1

On the other hand, of course,

[e_.p, elf] Thus [Zl' Zl*]

=

~ qi~i= if;

=

-if;.

0 and hence Zl is normal with respect to f.

Q. E. D.

6. 4. To gain information on cyclic elements in general it suffices by Theorem 6. 2 to focus attention on a single fixed cyclic element. We choose this element to be Zo, where Zo is given by (6.4.1) Here, of course, eo is given by (5.2. 2) . 14

164

1008

BERTRAM KOSTANT.

As in § 5. 2 let ao be the TDS spanned by X o, eo and fo. N ow let nk be the multiplicity of the irreducible 2k 1-dimensional representation 'lI"2k+l in the complete reduction of the adjoint representation of a o on go. By Corollary 5.2, nk = 0 if k is a half-integer and

+

(6.4.2) Now it follows from § 2. 5 ( d) and (5. 2. 4) that the kernel geo of ad eo is contained in~. On the other hand, it is obvious that geo n f) = o. Therefore, since Xo E f) is regular, it follows that the eigenvalues of ad Xo on geo are positive. This implies two things, (1) geo C n (Actually, we have already noted this fact-see proof of Corollary 5. 6) and (2) by § 2. 5 (d), no = o. That is, only the zero element is annihilated by ad a o• Indeed, this also follows from § 2. 5 (h) which asserts no = dim go- dim gl

=l-l =0.

For the present we direct our attention to the first fact, ge o C n. Let n* be the maximal Lie subalgebra of nilpotent elements given by

Then, of course, (6.4.3)

g=n* + f)+n

is a direct sum decomposition. We shall let p (resp. p*) be the projection p:

(resp. p*:

g~n

g~n*)

of g onto n (resp. n*) defined by the decomposition (6.4. 3) . Now consider the kernel gZo of ad Zoo given by

6.4A. Let po be the restriction of the projection to the subThen po (gZo) C geo and in fact

LEMMA

space gZo.

A relation between geo and gzo is

is a linear isomorphism of gZo onto geo.

165

COMPLEX SIMPLE LIE GROUP.

1009

Proof. It follows immediately from the decomposition (6.4.3) of 9 and (5.1.3) that [e-t/J, n] C n* q [e_1/I, q] C (e_1/I) [e_1/I, n*] = O.

+

Now let y E gZo be arbitrary.

Write

y=v +x+u, where vE n*, xE q, and uE n. Now by (6.4.1) and § 5.1

0= [zo,y] = [eo, v] [eo, x] [eo, u] (x, 1/1 ) e-ljt [e-ljt, u] .

(6.4.4)

+

+

+

+

Applying the projection p to both sides of (6.4.4) we obtain

[eo, x]

+ [eo,u] =0.

But then recalling the relation [gi, gj] C g»j, it is clear that [eo, x] = 0 and [eo, u] = O. Thus since u = p (y) it follows that p (gzo) C geo• It also follows that since geoC n, we must have

x=O.

(6.4.5) Thus y =v +u and

o=

[eo, v]

+ [eo, u] + [e-ljt, u] .

To show first that po is one-one, assume u = O. But this and the last expression imply that v E geo. But geoC n. Hence v = O. Thus Y = 0 which proves that po is an isomorphism into. But now, by Corollary 5. 3, dim geo= l. On the other hand, dim gZo > 1 (see, for example Theorem 5. 7). Since po is an isomorphism, this means dim gZo=l and also that po is onto. A major consequence of Lemma 6.4A is COROLLARY

6.4.

Q.E.D.

Cyclic elements are regular.

Proof. As above let Zo be defined by (6. 4. 1). It suffices to show Zo is regular. But by Lemma 6.3 Zo is semi-simple. Hence gZo contains a Cartan subalgebra. But, by Lemma 6.4, dim gZo = l. Hence Zo is regular. Corollary 6. 4 implies

166

BERTRAM KOSTANT.

1010 LEMMA 6.4B. subalgebra.

Let Zo be defined by (6.4.1).

Then gZo

~s

a Carlan

We will let ~' designate the Cartan subalgebra gZo. In the proof of Lemma 6.4 we considered the decomposition of any element y E ~' and showed that its projection x in ~ vanishes. (See (6.4.5». We state this as LEMMA 6.4C. Let Zo be defined by (6.4.1). Let p and p* be the projections of g on nand n* also defined as in § 6.4. Then the projection 1 - (p p*) of g on ~ vanishes on ~'.

+

Perhaps a simpler way of expressing Lemma 6.4C is to state that the Cartan subalgebras ~' and ~ are orthogonal to each other (with respect to B). 6.5. Now geo is invariant under ad Xo (see § 2. 5 (d) ). Let Ui, i = 1, 2,' . " l, be a basis of geo and also eigenvectors of ad Xo. Let ki' i = 1, 2,' . " l, be the corresponding eigenvalues. That is, Ui E gk" i = 1, 2, . . " l. We may regard the basis so ordered that ki < ki +1' At this stage we already know two of the ki' namely, the extreme ones kl and k!. We also know an inequality, kH k!.

<

LEMMA 6. 5A.

Let k i be defined as above. 1 = kl < k i < ... < k!_l

Then, where q =

< k! =

0

(tf!),

q.

Proof. As we noted in § 6.4 we must have kl > O. On the other hand, since eo E geo, it follows that kl = 1. But also e1fi E geo since tf! is the highest root. Thus [x o,e1fi] =qe1fi implies k =q. Since o(¢) < q for every ¢#tf! (see (5.3.4)), it also follows that kH < k l • Q. E. D. Taking the proof of Lemma 6. 5A into account we will choose U 1 = eo and u!=e1fi. N ow let di , i = 1, 2,' . " l, be the dimensions, in non-decreasing order, of the irreducible components occurring in the complete reduction of adjoint representation of a principal TDS (e.g. 0 0 ) on g. Appplying §2.5(d) the eigenvalues ki yield the dimensions di by the relation (6.5.1)

+

+

Lemma 6. 5A implies d1 = 3, dl = 2q 1 and d1r-l < 2q 1. N ow by Lemma 6. 4A there exists a unique basis Yl, Y2,' . " Yl of the Cartan subalgebra ~' with the property that p (Yi) = Ui. Define Vi by (6.5.2)

Yi=Ui+ Vi'

167

1011

COMPLEX SIMPLE LIE GROUP.

By Lemma 6.4C it follows that Vi E n* for i = 1, 2,· .. , l. Since U 1 = eo and since p(zo) = eo, it follows from (6.4.1) and Lemma 6.4A that V1 = e_tfJ and Y1 = zoo Now u! = etfJ. We shall have to know what V! is. LEMMA 6.5B. Let Ci and qi, i = 1, 2,· .. , l, be defined respectively by (5. 2. 2) and (6. 3. 1) . Then V! is the element given by I

L (qil Ci) e_a,.

V! =

(6.5.3)

.=1

Proof.

As in the proof of Lemma 6.3 I

[zo, etfJ

+ L (qJCi)e-aJ .=1

I

=

[e_tfJ, etfJ]

+ ~ qi [ea" e_a.] ;'=1

I

=-t/I

+ ~ qiai= O. i=l

I

Thus etfJ

+ ~ (qJ Ci) e_a, E ~'.

It follows immediately then from Lemma

;'=1

I

6.4A that

v!=~

(qJci)e-a,.

Q.E.D.

;'=1

6. 6. N ow it is clear (for example from the matrix representation of a TDS given in § 2. 5) that there exists an element A in the subgroup of G corresponding to 0 0 such that Axo = - Xo. But since Xo is regular, it is clear then that

In fact these relations are already contained in the more general fact

A:

gj~

g_j

for any - q < j < q. It follows then that we may interchange the roles of 6.+ and -6.+, nand n* and also gl and g-l in the results of §§ 6. 2-5. But then one sees that V! is principal nilpotent and that y! = u! V! is cyclic. But then by Corollary 6.4 it follows that gYI = ~'. Finally, applying Lemma 6. 4A we obtain

+

6. 6. We retain previous notation. Let p* 0 be the restriction of Then (1) V! is a principal nilpotent element, (2), P*o(~') C gVI and

LEMMA

p* to ~'.

in fact

168

1012

BERTRAM KOSTANT.

is isomorphism onto. In other words, the elements Vi, i = 1, 2,' . " l, form a basis of gVI. Also, the elements Vi commute with each other. (See Oorollary 5.8). It is obvious from (5. 2. 3) that VI is a nil-negative element of a principal S-triple containing Xo as neutral element or an nil-positive element of a principal S-triple containing - Xo as neutral element. Thus gVI is stable under ad Xo and since all principal S-triples are conjugate the eigenvalues of ad Xo on gVI are -lei, -lel-l,' . " -lel in non-decreasing order. The elements Vi are a basis of gVI by Lemma 6.6 but it is not at all clear yet that they are eigenvectors of ad Xo. 6.7. We now isolate a particular conjugate class in G, elements of which, will playa major role in the remainder of this paper. An element PEG will be called a principal element of G if there exists a principal regular element x E g (see § 5. 2) such that P can be written

P

(6.7.1)

=

Exp(27ri/s)x,

+

where s = q 1 and q, as usual, is the order of the highest root.p. Note that since x lies in a principal TDS, the principal element P lies in a subgroup of G corresponding to a principal TDS. (We shall make no use of the fact here but it can be shown that the number of principal regular elements x which satisfy (6.7.1) for a fixed principal element PEG is equal to order of the fundamental group of G). Various characterizations of principal elements will be given in §§ 8 and 9. Throughout the remainder of the paper we will let w be the primitive s root of unity defined by w = e27ri / B• Let Po be principle element in G defined by letting Po = exp (27ri/s) Xo. It is clear then that

for u E gj, - q < j < q. eigenvalues of Po and if

It follows then that wi, j = 0, 1,' . " q, are the Ui designates the corresponding eigenspaces, then

(6.7.2) where it is understood that g-8 denotes the zero subspace. It is an immediate consequence of (6. 7. 2) that Zo is an eigenvector of Po. In fact, (6.7.3)

Pozo =wZo.

But this clearly means that the Cartan sub algebra ~' = gZo is stable under Po.

169

COMPLEX SIMPLE LIE GROUP.

+

On the other hand, the elements Yi = Ui Let us apply Po to these basal elements.

1013

Vi, where Ui Egk" form a basis of 1)'. Then by § 6. 5

But the elements P OYi belong to 1)' and furthermore since, obviously, P oV. E n*, it follows that

But now if we apply Lemma 6.4A we must have

Thus % Ui and hence also Vi are contained in Uk,. But since Vi E n*, it follows from (6. 7. 2) that Vi E gk8' Thus not only did we prove that the Vi are eigenvectors of ad xo but more we obtain a duality relation among the integers lei' LEMMA Vi

E gk,-., i

Let

6. 7.

Vi,

i

=

1, 2,' . " l, be defined by (6. 5.2).

Then

= 1, 2,' . " l. Also the integers lei satisfy the following duality

law, s=lei

+ le =le Z

2

+le H =· . ·=le z + lei'

Proof. Only the second part of Lemma 6. 7 is not yet proved. By Lemma 6.6 the elements Vi are a basis of gV' and the first part of Lemma 6.7 asserts that Vi is an eigenvector of adxo with eigenvalue lei-so On the other hand, as we have seen, the eigenvalue;; of ad xo on gV' are, in non-decreasing order, -le z< -kH < .. ·<-ki • But these must be identical, and in the same order, as

ki-s < k 2 - s < ...
+

Subtracting termwise we obtain for all defined p, kp k Z+1-P = S. Q. E. D. Observe that the duality together with the inequality kH < kz implies the inequality ki < k 2 • In [3] Chevalley observed empirically a similar duality in the exponents (see [3], p. 24) of g. Later it will be proved that the integers ki are in fact these exponents and that the duality observed in the latter is the same as the duality just proved in the former. More than just a numerical coincidence the duality in the ki takes the following form. (Theorem 6.7 summarizes results which have already been proved.) THEOREM

6. 7.

Let c;, i

=

1, 2,' . ., l, be l arbitrary non-zero complex

170

1014

BERTRAM KOSTANT.

numbers. Let qi, i = 1, 2,' . " l, be the coefficients of the highest root 1ft relative to the simple positive roots CCi' Let !

eo = ~ Cie""

.=1

and

Then eo and eo are principal nilpotent elements in 9 so that geo and Ii;o are l-dimensional and commutative. !

Let

:1:0 =

~ £i,

where

~i,

i

=

1, 2,' . " l, is the basis of the Oar tan sub-

'£=1

~

algebra (:1: 0'

cp)

=

dual to the simple positive roots CCi, i = 1, 2,' . " l, so that a (cp) is the order of a root cpo For any non-zero integer j let gj= ~

(e",)

O(",)=j

be the eigenspace of ad:l:o for the eigenvalue j. Then geo and g'oo are each stable under ad:l:o and the eigenvalues of the restriction of ad:l:o to geo and g"O, respectively, are positive integers lei and the negative integers - lei, i = 1, 2,' . " l, where

, and q =

~ i=l

qi = a (1ft) and 2le 1

+ 1, 2le + 1,' 2

. " 2le!

+ 1 are

the dimensions

of the irreducible components occuring in the complete reduction of the representation of a principal TDS on g. Let and (cycle elements). Then Zo and Zo are regular, so that gZo and Ifo are Oartan subalgebras; and [zo, zo] = 0 so that gzo = g'o. Let Po, a principal element of G, be given by Po = Exp (271"i/ s) :1:0, where s = q 1. W rite ~' for gZo = g
+

Yi=Ui+ Vi,

171

COMPLEX SIMPLE LIE GROUP.

where the Moreover, relation

1015

form a basis of geo and the Vi form a basis of geo, i = 1, 2,' . . ,l. Ui Egk, and Vi E gk,-s so that the integers k i satisfy the duality

Ui

Finally we can choose Y1 =

Zo

and Yl =

zoo

6. 8. Among other things, Theorem 6. 7 asserts that the integers k i may be found by considering the eigenvalues, w k ', of the restriction of the principal element Po to fJ'. Theorem 6.8 below asserts that without knowledge of ~' or the restriction Po I ~' we can determine the integers k i directly by considering multiplicities of eigenvalues of a principal element with respect to its action on g. In the notation of § 6. 4 and Theorem 6.8 this means we can express nk in terms of Sk. THEOREM 6.8. Let P be a principal element of G. Let Sk, 7c = 0,1, ... ,q, be the multiplicity of the eigenvalue w k of P (In the notation of § 6. 7, Sle = dim gk gk-8) and as in § 6.4 let nk be the multiplicity of 7T2k+1 in the adjoint representation a principal TDS on g. Then

+

(6.8.1)

In other words, let ~1 be a Cartan subalgebra which is stable under P and is such that the eigenvalues of P I ~1 are w k ', i = 1, 2,' .. , l (such a Cartan subalgebra exists by Theorem 6.7). Then if fi1 is the B-orthocomplement to ~1' fi1 is stable under P and the eigenvalues of P I fi1 are w k , k = 0,1, ... , q. Furthermore each eigenvalue of P I fi1 occurs with multiplicity l. Proof. Since any element in G is orthogonal with respect to B, it follows that fi1 is stable under P. It suffices then to assume P = Po and ~1 = ~ to prove Theorem 6. 8 for this case. Now by § 2. 5 (h)

(6.8.2) Since no = 0 upon summing (6. 8. 2) we obtain for 0 < k < q (6.8.3)

k-1 dim go-dim gk = ~ nj. ;=1

But by § 2. 5 (f) q

(6.8.4)

dim g8-1e =

~ nj. j=8-k

But by the duality in Theorem 6. 7 ns-j = nj. But then subtracting (6. 8. 3) from (6.8.4) and recalling that dim g8-1e = dim gk-s and dim go = l it followR that Q.E.D.

172

1016

BERTRAM KOSTANT. COROLLARY

s= q

6. 8.

Let r be the number of positive roots of g. order of the highest root.p. Then

+ 1, where q is the

Let

ls=2r. Proof. This immediately follows from the second part of Theorem 6. 8 since dim ml = 2r. Q. E. D. Since n = l 2r note that it also proved that l divides n. In fact n=l(s+l).

+

7.

The notion of the apposition of two Cartan sub algebras.

1. Let 1)1 be any Cartan subalgebra of g. Let HI be the corresponding subgroup of G. Let N (HI) be the normalizer of HI in G. It is obvious that A E N(Hd if and only if 1)1 is stable under A. Let WI be the subgroup of endomorphisms of 1)1 induced by restricting the elements of N (H 1) to f)1. Then the restriction defines a homomorphism

,: N(Hd

~

WI

of N (HI) onto WI. The group WI is finite and is called the Weyl group with respect to f)1. It is well known that the kernel of , is HI so that N(Hl)jHl ~WI. If u E WI and A E N(H 1 ) and A E N(Hd is such that ,(A) =u, then A will be called an extension of u. It is clear then that the most general extension of u is of the form AlA, where Al E HI. Let ~1 be the set of roots with respect to f)I. It is well known that ~1 is stable under WI. For each root cp E ~1 let Rrp be the "refiection" of f)1 defined by (7. 1. 1)

Rrpy = y -

(2 ( cp, y) /

(cp, cp) ) cp

One knows that Rrp E WI for all cp E ~1 and that WI is generated by these refiections (See [12], Theoreme 1, p. 16-05). 7.1. Let u E WI. Let 11. be an eigenvalue of u. Then 11. is a primitive j-th root of unity for some j and every other primitive j-th root of unity occurs as an eigenvalue of u. LEMMA

Proof. Since ~1 is stable under u, there exists a basis of f)1 (e. g. a set of simple positive roots) with respect to which u is represented by a matrix with rational coefficients. It follows then that the characteristic polynomial of u also has rational coefficients. N ow since u has finite order, 11. must be a primitive j-th root of unity for some integer j. It folows then that the characteristic polynomial of u is a product of cyclotomic polynomials and

173

1017

COMPLEX SIMPLE LIE GROUP.

hence every primitive j-th root of unity is a root of the characteristic polynomial. Q. E. D. 7.2. Let Ul and n Q, the set of Q-simple roots with respect to D, be defined, respectively, as in § 6. 7 and § 6. 2. It is clear that U 1 is the set of all elements y E g of the form Y= ~ afJefJ fJeITQ

We know that when all the coefficients ap are different from zero, y is cyclic, and hence also regular. It is interesting that in case one of the coefficients is zero, not only does y cease to be regular, it is in fact nilpotent. LEMMA

7.2.

Let rrQ be as in § 6. 2.

Let y E g be of the form

y= ~ apep. peITQ

That is, in the notation of § 6. 7, assume y satisfies P oy = wy. of the coefficients ap is zero, y is nilpotent.

i

=

Then if one

Proof. Assume one of the coefficients is zero. In fact, let f3i' 1, 2,' . " l 1, be the elements of rrQ. Without loss assume ap,,, = O.

+

I

N ow upon writing t/J =

~

q,lX, we know that the coefficients q, are all

i=l

non-zero (See (5.3.3)). It follows immediately then that the roots f3" f32,' . " f31 form a basis of g. But if we use this basis to define a lexicographical ordering in g# and set 3.+ equal to the set of positive roots relative to this ordering, it follows that f3i E,3.+ for i = 1, 2,' . " l. But then n = ~ (e",) is a Lie algebra of nilpotent elements and y E it. Hence y is ¢

..0.+

Q. E. D.

nilpotent. 7.3.

Applying Lemma 6.2 we obtain the corollary

LEMMA 7.3. Let y E U , . Let H be the subgroup of G corresponding to Then y is either nilpotent or else there exists A E H and a non-zero scalar A such that

g.

y=AAzO'

For any element A E G let gA be the set of fixed vectors of A. An element A EGis then said to be regular if (1) A is semi-simple (that is, A is diagonalizable) and (2) gA is a Cartan subalgebra. It follows from Theorem 6.8 that a principal element PEG is regular. Now a special relation, to be defined below, exists between the pair of

174

1018

BERTRAM KOSTANT.

orthogonal Cartan subalgebras ~ and ~' (See Theorem 6. 7) . We will show that this relation defines ~ and ~' uniquely up to conjugacy. Let ~l and ~'l be two Cartan subalgebras of g. We will then say that ~'l is in appostition to ~l with respect to a principal element PEG if (1) gP = ~l and, (2) ~' 1 is stable under P and the set of eigenvalues of P I ~' 1 includes a primitive s root of unity. (Observe that the number s has special significance for a principal element; the order of a principal element in G equals s.) By Theorem 6.7 it is clear that ~' is in apposition to ~ with respect to Po. 7.3. Assume that the Cartan subalgebra ~'l is in apposition with respect to the principal element P 1 and that ~' 2 is in apposition ~2 with respect to P 2 • Then there exists A E G such that AP1 A-l=P2 , A~l = ~2' and A~'l = ~'2. THEOREM

to to

~l

Proof. It suffices to assume P 1 = P 0, ~l = ~ and ~' 1 = ~'. N ow all principal elements in G are conjugate to each other. Hence there exists Al E G such that A 1 P oA l-l=P 2 • Since gP2=~2' it is obvious that Al~=~2. Thus it also suffices to assume ~2 = ~ and P 2 = Po. But now ~' 2 is stable under Po and Po I ~'2 has a primitive s root of unity as an eigenvalue. By Lemma 7.1 w is also an eigenvalue of Pol ~' 2. Let y E ~' 2, YoF 0, be a corresponding eigenvector. Then in the notation of § 6. 7, Y E U l • But since y is contained in a Cartan subalgebra, y is not nilpotent. Hence by Lemma 7.3 there exists A E G and a non-zero scalar A such that y = AAz o• But then by Corollary 6.4 Y is regular and hence A~' = ~'2. Q. E. D.

8.

The Coxeter-Killing transformation.

1. An element y' E W will be called a transformation of Coxeter-Killing if it can be put in the form

where Il'i, i = 1, 2,' . " l, are the simple positive roots, in some order, relative to some Weyl chamber in ~#. In recent years this transformation has been studied by Coxeter. Relationships between the eigenvalues of this transformation and the eigenvalues of the matrix of Cartan integers have been established. (See [5]). Also one knows that all the transformations of Coxeter-Killing in W form a conjugate class in W. Let y = R a,R a2 · •. R a ,. An integral valued function on the Weyl group,

175

COMPLEX SIMPLE LIE GROUP.

1019

which plays an important role in the cohomolgy theory of complex homogeneous spaces and also of n is the function N «J) which assigns to each E W the number of positive roots which go over into negative roots under the action of (J". The knowledge of N (y) will play an essential role in proving that the order of y is s. (J

THEOREM 8.1. Let y be the transformation of Coxeter-Killing defined Then N (y) = l. Furthermore if CPt, by letting y = R""R",.· .. R"". i = 1, 2,· . ., l, are the positive roots which change sign under y then the roots CPt form a basis of f).

Proof.

Let CPi, i

=

1, 2,· . . ,l, be defined by

It follows easily from (7. 1. 1) that

cpt =

Oli

+1>1 ~

CijOlj.

Since the coefficient of Ol, in CPt is one, it follows that cpt is positive. Furthermore, since the coefficient of Olj in cpt for j < i is zero, it is clear that CPt are linearly independent and hence form a basis of f). But clearly and hence

Since the coefficient of Ol, in y (cpt) is minus one, CPi is a positive root which changes sign under y. Thus N (y) > l. N ow it is well known that N (R",,) = 1. In :fact, Ol. is the only positive root which changes sign under R"". This is clear since only the coefficient of Ol, in cP, :for any cP E ~, is affected by R"". N ow assume cP is a positive root which changes sign under y. Since y(cp) E ~-, there clearly exists a maximal value i such that

That is, R""+l· .. R""cp E ~+.

It :follows then that

This, however, means that cp = CPl. Thus N (y) = l. Q. E. D. Now it is well known that one is not an eigenvalue o:f a Coxeter-Killing trans:formation. A proo:f of this resting on the work of Coxeter is given, :for

176

BERTRAM KOSTANT.

1020

example, in [4]. (See [4], p. 352). However, since a direct proof of this fact, making use of Theorem 8.1, can be given we shall include it. LEMMA 8.1. Let y be the Coxeter-Killing transformation given as in § 8. 1. Then one is not an eigenvalue of y. Let £i, i = 1, 2,· .. , l, be the basis of Clearly

Proof.

q given

as in § 5. 1.

Let

l. = 2£i/ (ex., ex.).

yl. = R a, · .. Rail. l.

=

+ Ra, · .. Ral+l (- exl)

by (7. 1. 1). On the other hand, it is clear from the definition of CPo in the proof of Theorem 8.1 that R Ra,•, (- ex.) = y (cp.). Thus we have

a, · ..

yl. = l.

+ y(cp.).

z Assume now that x E q is fixed under y. Write x = ~ ail..

Then

.=1

x=yX z

=

~ a.;(l.

0=1

+ 1'( CPo))

Z

=x+ ~ai'Y(cp.). i=l

z

Thus

~ .=1

acy (cpt)

=

o.

However, by Theorem 8. 1 the vectors CP. and hence

y (CPi) are linearly independent. x=o.

8. 2.

Thus a. = 0, i

=

1, 2,. .. , l, and hence Q.E.D.

Lemma 8. 1 is needed solely to prove

LEMMA 8. 2. Let r. CA, i = 1,2,· .. , L, be the orbits in A under the action of y. Then for i = 1, 2,· .. , L

Proof. This follows immediately from Lemma 8. 1 since the sum of the roots in any orbit of y is obviously left fixed by y. Q. E. D. The following theorem is proved in [4] THEOREM 8.2 (Coleman). Let h be the order of the Coxeter-Killing transformation y defined as in § 8.1. Let v = e27ri / h • Then there exists a regular eigenvector Zl of I' whose corresponding eigenvalue is v. Coleman also observed and used in [4] the following consequence of Theorem 8. 2. We repeat his proof.

177

1021

COMPLEX SIMPLE LIE GROUP.

8. 2. As in Lemma 8. 2 let r, C a, i = 1, 2,· . ., L, denote the distinct orbits of "Y with respect to its action on the set of roots a. Then each orbit r. contains exactly h roots so that in particular hL = 2r, where 2r is the total number of roots. COROLLARY

Fro of. Let cp E a. Assume ymcp = cpo It suffices to prove that h divides m. Now, where Zl is given as in Theorem 8. 2, (Zl' cp)

(Zl' ymcp) (y-mZl' cp ) y-m (Zl' cp) •

= = =

But since Zl is regular, (Zl' cp) # 0 and hence divides m.

y-m =

1.

This implies h Q. E. D.

8.3. Now one knows that the Poincare polynomial Fa(t) can be put in the form

Fa(t)

=

(1

+t

2mt+1 )

(1

+

t2m2+1) • • •

(1

+t

2ml +1 ) ,

where the mi, i = 1, 2,· .. , I, are positive integers in non-decreasing order. The integers mi (sometimes m. 1) are called the exponents of g (or W as in [4J). When the values of the exponents for the simple exceptional Lie algebras were announced by Chevalley at the International Congress at Cambridge in 1950, Coxeter recognized a rather remarkable coincidence. He observed (1) that in all cases hI = 2r, so that in our notation h = s, and hence y=ro (See Corollary 6.8), (2) mi
+

+

+

of y.

8. 4. N ow let y be defined as in § 8. 1. Let A'Y E G be any extension Let ri C 6., i = 1, 2,· .. , L, be defined as in § 8. 2 and let

(8.4.1)

178

1022

BERTRAM KOSTANT.

By Corollary 8. 2 it is clear that each subspace O. is of dimension h. Furthermore, since for any cf> E ,A, it is clear that

is a direct sum decomposition of 9 into subspaces which are stable under A'Y' Now since "'/' is the identity element of W, we can define the scalar A~, for any root cf>, by the relation (A'}ye~ = A~e~.

(8. 4. 2)

By applying A'Y to both sides of (8.4. 2) it is clear that A~ = A'Y~ for all cf>. Now since A./, reduces to the identity on g, we can write A./, = Exp x for some x E g. It follows that A~ = e(IIJ.¢) for any cf> E A. But then /&-1

(A~)"=n A-y'~ ~=O

=

e(IIJ.¢+'Y¢+ .•• +'Y"-1¢)

=1 by Lemma 8.2. Thus A~ is an h root of unity. This shows that A./" = 1. We can now prove that h = s and hence that L = l. More than this we have THEOREM

subalgebra

g.

8.4. Let'Y be a Coxeter-Killing transformation on the Cartan

+ ~ q., where the integers J

Let h be the order of 'Y' Then h = 1

i=1

q. are the coefficients of the highest root relative to a basis of simple positive roots. That is, h = s. Moreover, hl = 21', where l' is the number of positive roots so that there are l distinct orbits r. C A, i = 1, 2,' . " l, in A under the action of 'Y and each orbit contains h roots. Furthermore, if we take 'Y = Ra,Ra.· .. R a, and let cf>i' i = 1, 2,' . " l, be the positive roots which change sign under 'Y (see Theorem 8. 1), then we can choose an ordering of the orbits so that cf>i E r i , i = 1, 2,' . " l. Now let A'Y E G be any extension of 'Y' Then A./, = 1. That is, A'Yhe~ = e~ for any cf> E A so that we can renormalize the root vectors e.p in such a way that A'Y'e~ = e'Y'~

for any cf> E A and any integer i. Now let w. E g, i = 1, 2,' . " l, be defined by

w.=

~ e~

¢Er,

179

COMPLEX SIMI'],E UE

1023

(movl'.

and let ~ be the subspace spanned by the elements Wi. Then 1) is a Cartan subalgebra of 9 (so that the elements Wi are semisimple and commute with each other). Furthermore, ~ = gA~. Proof. Let y, A-y and A~ for rp E ~ be as previously defined. It has been = 1. In particular A-y is semisimple. But if A EGis semishown that simple it is well known that gA contains a Cartan subalgebra of 9 (See e. g. [8J). For A = A'Y the proof is somewhat more direct since A'Y' having finite order, lies in a maximal compact subgroup of G. But then A'Y = Exp x for some x in a compact form of g. We could then apply § 6. 1 (d) ). Let 1) be a Cartan sub algebra of 9 contained in gA~. N ow let Ii;, i = 1, 2,· . ., L, be defined by (8. 4. 1) . Consider the decomposition

kr

L

g=~+LIii. ;=1

Obviously gA~ = ~

L

n gA~ + ~ Iii n gA~ i=l

(8.4.3)

by Lemma 8. 1. N ow let rp E rio As we have already noted A~ = A'Y'~ for any integer i. Thus A-yh reduces to the scalar A~ on the space Iii. But then if A~ =1= 1, A'Yh has no non-zero fixed vectors in Iii and hence certainly Iii n gA~ = O. On the other hand, if A~ = 1, then clearly Iii n gA~ is the one dimensional subspace 1&-1

spanned by

~ A'Yie~.

Thus if L1 is the number of integers i, 1 < i < L, such

i=O

that A~ = 1 for all rp E r i , it follows from (8. 4. 3) that Ll = dim gA~. In fact, since ~ C gA~, we then have

L > L1 = dim gA'l' > 1 and hence in particular L > l. Now we assert that for any i, r i n~+ and r i n fJ.- are both non-empty. Indeed assume, without loss, that r i n fJ.- is empty. Then L rp, in the ¢€r,

lexicographical order of ~#, must be strictly positive and hence cannot vanish. This contradicts Lemma 8. 1 and hence the assertion is proved. N ow since r i n fJ.+ and ri n fJ.- are non-empty, it is clear that there exists a root rp E r i n ~+ such that yrp E r i n fJ.-. That is, each orbit r i contains at least one positive root which changes sign under y. But then if we apply Theorem 8. 1 it follows obviously that L < l. Thus L = Ll = dim gA~ = l. That is, A~ = 1 for all