The Capelli identity, the double commutant theorem, and multiplicity-free actions

Math. Ann. 290, 565-619 (1991)

i

9 Springer-Verlag1991

The Capelli identity, the double commutant theorem, and multiplicity-free actions Roger Howe t'* and T6ru Umeda 2"** t Department of Mathematics, Yale University, Box 2155 Yale Station,

New Haven, CT 06520, USA 2 Department of Mathematics,Facultyof Science, Kyoto University, Kyoto 606, Japan Received October 23, 1990; in revisedform April 2, 1991 Introduction

0. The Capelli identity [Cal-3; W, p. 39] is one of the most celebrated and useful formulas of classical invariant theory [W; D; CL; Z]. The double commutant theorem [W, p. 91] is likewise a basic result in the general theory of associative algebras. Both play key roles in Weyl's book: The classical groups. The main purpose of this paper is to demonstrate a close connection between the two, in the context of multiplicity-free actions EKJ of groups on vector spaces. The focus of our discussion will be the structure of the differential operators which commute with a multiplicity-free action. Applications include a new derivation of some formulas of Shimura [Shl ] and Rubenthaler and Schiffmann [RS] for b-functions associated to Hermitian symmetric spaces, and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin. We also give a detailed discussion of certain aspects of multiplicity-free representations. Here is an overview of the contents of the paper. In Sect. 1 we review the classical Capelli identities. We observe how their existence is predicted by a double commutant result (1.6) and (1.8), and we show how they can be used to compute b-functions. We remark that Capelli understood that his operators generate the center of the algebra generated by the polarization operators (see [B1, p. 77]). Also Capelli's motivation in introducing his operators was to "explain" a formula of Cayley - essentially the computation of a b-function. Capelli's point of view was remarkably modern and structuralist, in certain ways more modern even than that of Weyl. Sections 2 through 9 provide a conceptual context for understanding the Capelli identities as a feature of multiplicity-free actions. They contain a general discussion of the structure of ~ G , the algebra of polynomial coefficient differential operators which commute with a given group G of linear transformations. The results of the discussion are summarized in Theorem 9.1, which says in particular that the polynomial coefficient differential operators commuting with a * Partially supported by NSF grant ~DMS-8807336 ** Partially supported by a FellowshipProgram of Ministryof Education of Japan

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given multiplicity-free group action is a polynomial algebra on a canonically defined set of generators. In the case of the action GL. x GL. on the n x n matrices, these generators are precisely the CapeUi operators. Theorem 9.1 also describes seven different algebras, all more or less naturally isomorphic to each other, any of which can be used to compute ~ ~ On the basis of Sects. 2-9, we formulate in Sect. 10 two "Capelli problems" for multiplicity-free actions. We point out that the "abstract Capelli problem" amounts to a double commutant theorem. The long Sect. 11 is devoted to studying these problems for the list of irreducible multiplicity-free actions given by Kac [K]. We give more or less explicit analogues of the Capelli identities for most of the actions on Kac's list. Perhaps the most interesting example is the action of GL. on the skew-symmetric n x n matrices. Sections 12-14 discuss some algebro-geometric aspects of multiplicity-free actions naturally related to our main investigation. We show that, in a multiplicity-free action by a group G, the G-orbits correspond canonically to certain subsets of the generating set o f ~ ~ and that this correspondence imposes a weak partial-order on the generating set. We compute this correspondence explicitly in the various examples, thus obtaining an explicit description of the ideal of polynomials vanishing on any G-orbit. For the cases of GL. x GL. acting on the n x n matrices, or GL. acting on the symmetric or antisymmetric matrices, these are classically studied "determinantal ideals" (see rR] and references therein). In Sect. 15 we give a table summarizing the results for the various examples. In the Appendix we give an efficient proof that the classical Capelli operator is in the center of the enveloping algebra of gl., and show that a similar construction is valid for the orthogonal Lie algebra. 1. To introduce our main themes, we discuss certain aspects of the standard Capelli identity. Let M,(C) = M. be the space of complex n x n matrices. A typical element of M,(C) is a matrix

Its1.. t~2 ... t~..1

"=!

'1"

Lt., ............

't.:. 1

We will use the entries t o of T as coordinates on M.. The basis with respect to " 1, each of which the t~j are coordinates is the set of standard matrix units {E u}~,r which has exactly one non-zero entry, which is 1. Thus expression (1.1) is equivalent to the expansion

T= ~ t~jE~j.

(1.2)

i,j=l

The group GL.(C)=GL, can act on M. by left multiplication or by right multiplication. These actions commute with each other. Both actions give rise to actions on the algebra ~(M.) of polynomials by the usual formulas: (1.3a)

L(g)(P)(T)=P(g-IT)

(geGL.,TeM.,Pe~'(M,))

for the left action, and (1.3b)

R(g)P(T)= P(Tg)

The Capelli identity

567

for the right action. Differentiation of L and R along one-parameter subgroups gives rise to actions of the Lie algebra gl.--- M. of GL. on ~'(M.) via vector fields with linear coefficients. Explicitly, we have

(1.4)

L(E0=- ~ tizd., l=~ R ( E 0 = Y, tuOu. /=1

Here 0 dq= &U indicates partial differentiation with respect to t~j. Because of their origin, we know the operators L(Eu) commute with the operators R(Eu). The actions L and R of gl. on ~(M,) extend uniquely to homomorphisms

(1.5)

L: ~(91.)-. ~ ( M . ) R: q/(9I,)--,~(M.)

of the universal enveloping algebra ~(gl~) of 91. to the algebra ~@(M.) of polynomial coefficient differential operators on ~(M.). Clearly the two images L(q/(gl.)) and R(q/(flI.)) commute with one another. As a special case of the version of Classical Invariant Theory described in [H1], we know that L(~d(gl.)) and R(q/(flt~)) are each the full commutant of the other inside ~ ( M . ) . It follows that, if ~q/(flI.) is the center of q/(91.), then (1.6)

L(~q/(gl.)) = L(ql(gI.)) n R(q/(g[.)) = R(~'(91.)).

This fact has the following consequence. We may combine the mutually commuting left and right actions GL. into a joint action, denoted L x R, of GL~ x GL.. It is obvious that for the differentiated form of this action we have (1.7a)

(L x R)(gl,@ gI.)= L(gl.)+ R(91,).

Hence (1.7b)

(L x R)(q/(gI.~fll.)) = L(~(gl,))- R(q/(gl.)).

We see from 1.7b) that using the joint GL. x GL, action allows us to reformulate (1.6) as (1.8)

(L x R)(q/(9I, ~9 gl,))' = ~e((L x R)(q/(gl,~ gl,)))=/_,(~rq/(gl,)) = R(Lrq/(91,)) 9

Here the' indicates commutant inside ~ ( M , ) and ~ placed in front of the symbol for an algebra indicates the center of the algebra (i.e., the commutant of the whole algebra in itself). Equation (1.8) say that any polynomial coefficient differential operator on ~(M,) which commutes with both the left and fight actions of GL, must come from ~q/(91,), via either L or R. Thus, if we have given to us a differential operator d that commutes with left and fight multipliation by GL., we are guaranteed by (1.8) that we can find an element za in ~Yq/(91.)such that A = R(za). For example, consider the polynomial function detT=det{tzj} on M.. This is an eigenfunction for L x R: (1.9)

(L x R)(g~, gz) (det T) = (det g 0 - ~(det g2)det T

(g, ~ GLn).

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R. Howe and T. Umeda

Similarly, the differential operator (the "Cayley f2-process" [W, p. 42]) (1,10)

f2 = d(det T) = det {0o}

formed by taking determinant of the partial derivatives d~j, is also an eigenvector for the action (L x R)* of GL. x GL. on ~(M.), the algebra of constant coefficient differential operators. When the two are multiplied together we obtain an operator (1,11)

C~= C = (det T)O

which commutes with GL,, x GL,,, and for which there must be a corresponding Zo The challenge of explicitly describing zc was met by Capelli [Ca l ], who found the famous identity (1.12)

(det T)f2 = det {R(E,9 + 5~j(n- j ) } .

Here the determinant in the right hand side means the alternating sum of products of entries, one from each row and column, the order of the factors in the product being the same as the order of the columns the factors come from (cf. [W, p. 40; H1, p. 564]). Thus, because the existence, though not the exact form, of (1.12) is predicted by (1.8), it makes sense to refer to (1.8) as an abstract Capelli identity. The concrete Capelli identity (1.12) has numerous uses (see for example [W; D; CL; Z; S], ...). Here we would like to illustrate how a slight extension of it can be used to compute the b-functions studied in [Shl] (see also [RS]) for the case of M,. To discuss the b-functions, we need to recall the decomposition of ~(M,) into irreducible subspaces for GL. x GL.. Before we describe this, it is convenient to modify slightly the action ofGL. x GL.. This will not change in any important way the picture we will describe, but it will make certain parts slightly simpler to talk about. Instead of the action L described by (1.3), we will use the composition of this action with the automorphism g__,(gt)-1, where gt is the matrix transpose of g. Thus we consider the action (1.3')

E(g) (P) (T) = p(gtT).

This has the effect on gI. of replacing x by - x t. Thus

(1.43

/:(E~j) = - L(E~)= ~ t~j~. 1=1

Under the action E • R of GL. • GL., the polynomials break up into a multiplicity-free sum of irreducible GL. • GL. representations. Specifically (1.13)

~(M.) _~ y. q.o| D

Here we are using the description of representations of GL, in terms of Young Diagrams or highest weights. The symbol D here denotes a decreasing sequence of non-negative integers: (1.14)

D=(al, a2..... a,)

(at>a~+l, at~Z+).

The symbol #o denotes the irreducible representation of GL. with highest weight parametrized by D with respect to standard coordinates on the diagonal torus; see [W'J or [H4] for a more detailed explanation. The copy of GL. acting via L' acts on the first factor in the tensor product 0~~174~ and the copy acting via R acts on the

The Capelli identity

569

second factor. One can easily describe the set of GLn x GLn highest weight vectors

itl~., tl 2

in ~(M~). Set (1.15)

~k=det

~1

tl.k1

...

.......

.

Ltk1 . . . . . . . . . . . "t~'kj Then the highest weight vector which generates the subspace of ~(M,) on which GL~ x GL~ acts by Q~o | o is (1.16)

4,o_,al-a2~,a2o3" " u..... 1- ~o,a, --/'1 i'2 ~'n 9

/

Here a i are the components of D, as in (1.14). Now we can introduce the b-functions of Shimura. Our discussion will be superficially different from Shimura's, but can easily be seen to be equivalent. Consider the differential operators dl I "''

(1.17)

dlk

d(~k) = det Ld~ 1

dual to the 7k, and also the minors tk+lk+l

(1.17)

tn~

...

complementary to the YR"Set (1.18)

ii]

"'" tkn

~k=det

C~(yD)= fi C~(yk)a~-~+~,

t

3~D= fi )y~-,~+l.

k=l

k=l

We use the common notation

(1.19)

IDl= E a~. k_>0

General considerations tell us that

8(7~ (det T ~)= flD(S)(det T) ~-"~~

(1.20)

where flo(s) is an appropriate polynomial. However, to determine the exact form of flo requires a calculation9 We will show how to use the Capelli identity to do the calculation. Actually, we need the more general analogs of(1.12) given in [H1, Sect. 4f] (in fact these were known to Capelli ECa3]; see Sect. 11.1 and Appendix for a proof). For 1, J subsets of N = {1, 2, ..., n}, let T/I be the submatrix of T formed from those t~j such that i e I and j ~ J, and let O(T)u be the matrix obtained by replacing each t o in T~ by a o. For ! c___ N, of cardinality k, denote by H~ the matrix

-R(Ei~i) + k - 1 R(Eizil ) HI_ ~

R(Ei~i) ... R(Eilik)l R(Ei2i2) + k - 2 ... R(Ei~ J [

.

.

.

[ "

R(Ei~i)

...

R(Ei~ik) j

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R. Howe and T. Umeda

The order k Capelli identity says (1.21)

Z det(T~j) det(d(Trj)) = Y. det(H,). l,J

I

Here I (on both sides) and J (on the left) vary over all subsets of N of size k. The non-commutative determinant d e t H I is defined in the same way as in (1.12). Consider now the problem of computing/~a- We claim it is enough to know tip for D = D k, where D~ has the form ai = 1 for i < k, as = 0 for i > k, i.e., when ~D= ~:k.To see this, observe that the variables in ~ are completely distinct from the variables in ~6 as long as k < I. Hence 0(Yr) ((det T)St~6 ~ ) = (0(~6)(det TS))G>=[Ik~z). Thus if we compute 0(~D)(det T ~) by applying the 0(76) in order, starting with k large, we find the formula fig(s) = I~I "~-fik+~flD~(s--(ak +, + j-- 1)).

(1.22)

k=l

j=l

Thus we only need to compute the basic polynomials flo~. We do this using the Capelli identities (1.21). The specialization of (1.20) to the case D =Dk is

O(rk)(det T ~)= fiDe(s) (det T) ~- 1 ~k" Because d e t T is invariant under SL.(R), we can conjugate this equation by permutations of the variables to obtain det 0(Tu) (det T ~)= _+fiDe(s)(det T) ' - l det( TN- i. s - J) for any subsets I, J of N of cardinality k. Multiplying by det T~j, summing over I and J, and using the rule for expanding a determinant as a sum of products of complementary minors, we find that

(~jdet(T~z)det(O(Tzz)))(detT~):(:)flD~(S) detT~. On the other hand, det T is annihilated by R(E~9 if i~:j. It follows that, if we apply one of the terms det(//1) from the right side of(1.21) to det T ~, the only one of the k! k products which does not annilate det T ~ is the diagonal one lrI (R(E~, 0 + k-c). r

Since det T" is homogeneous of degree s in each of its columns, it follows that

(0, R(Ei,~) + k - c,), det T ~)= 1-I' (s + k - c) = C

r

l-I (s + c). c=O

Summing over all I of cardinality k, and using (1.21), we conclude 6--1

(1.23)

flo~(s) = I-I (s + c). r

Generalities 2. Beginning in this section, we wish to put the facts and computations of Sect. 1 in a more general context, that of multiplicity-free actions. Let G be a connected

The Capelli identity

571

reductive complex algebraic group acting on a vector space V. We say G is multiplicity-free if the natural action of G on the algebra ~(V) of polynomial functions on V is multiplicity free as a representation of G, i.e., each irreducible representation of G occurs at most once in ~(V). From (1.13), we see that the action L x R of GL n x GL n on Mn is multiplicity-free. The irreducible multiplicity-free actions have been classified by Kac I-K]. See Sect. 11.0. Recall that Sato [S] (see also I-SKI) calls a vector space V with a group G acting on it prehomogeneous if G has a Zariski open (hence dense) orbit in V. Multiplicityfree actions are all prehomogeneous. In fact, according to [Se; VK], the action of G on V will be multiplicity-free if and only if a Borel subgroup B of G acts prehomogeneously on V. It then follows from I-S, Theorem 1] or [SK, Sect. 4, Corollary 6] that the B-eigenvectors in ~(V), which are of course exactly the highest-weight vectors of the irreducible representations of G in ~(V), have a very simple structure. Let Q be a B-eigenvector, with eigencharacter ~, so that (2.1)

Q(b-1 v) = tp(b)Q(v)

(b E B, v ~ V)

for a suitable character tp of B. Then Q is in fact determined up to multiples by yd. (This is an easy consequence of the assumption that B has a dense orbit, see t-S, Proposition 20)] or [-SK, Sect. 4, Proposition 3].) We write Q~ to denote a B-eigenfunction whose eigencharacter is ~p. Clearly, up to multiples,

G1Q~2=Q~,~2 so that the set of ~v such that Q~ exists forms a semigroup. Let us denote this semigroup by/~+(V). (It will be a subsemigroup of the cone/~+ of dominant weights of B.) Recall that an element of a semigroup is primitive if it is not expressible as a product of two elements of the semigroup. If ~v=~p:p2, then Q~= Q~,Q~,2. Hence, if ~p is not a primitive element of our semigroup, the polynomial Q~ cannot be prime. On the other hand, since B is connected, one can see by unique factorization [L, Chap. V, Sect. 6] that the prime factors of any Q~ must be B-eigenvectors. This combined with the uniqueness of the Q~ allows one to conclude that there are a finite number of primitive characters {~pj}t ~i~- of B, such that corresponding eigenvectors Q. are prime polynomials, such that--ff+ (V) is the free abelian semigroup generatec~ J by the ~pj. Thus a general element of the semigroup has the form

(2.21)

~= IrI ~

(cj~Z +)

j=l

with the cj's uniquely determined, and the corresponding Q~ has factorization (2.2b)

Q~ = I~I (Q~yJ. j=l

The polynomials Q,~.,,are unique up to multiples and ordering. Since the characters ~j are linearly independent as elements of the character group of B, we can use the action of B to multiply the Q~.J by arbitrary non-zero scalars. Hence the Q~03 are . . . unique up to the action of B and ordenng. In fact, as we will see Sect. 12, there is a natural order relation on the ~pj,and in many cases, this ordering is actually a total ordering. In the case of the action of GL,, x GL,, on M,, discussed in Sect. 1, the Q~j are the Yk of (1.15).

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As we have remarked, the highest weight theory for representations of G [B2] implies that the transforms by G of each B eigenvector Q, span an irreducible G-invariant subspace of 2(V), and all irreducible G-invariant subspaces arise in this fashion. Thus we can write (2.3)

2(V)= ~,~(v) Y~'

where Y~is the irreducible G-invariant space containing Q~,.We remark in passing that if ~p is factored as in (2.2) a), then Y~,~_2Z(V), the space of polynomials homogeneous of degree l, where l= ~ c~degQ~j. j=l

3. As advertised in the Introduction, our main interest is in the structure of the polynomial coefficient differential operators commuting with G. These can be looked at in several different ways, and we will spend some time detailing the relation between different viewpoints. This discussion is summarized in Theorem 9.1. The equivalence of these various viewpoints is probably known to experts, and some are implicit in the literature [J: Shl ; RS] but the full picture is sufficiently rich that an explicit treatment seemed worthwhile. To begin we recall some well-known facts about differential operators. A fuller discussion of these facts can be found in various places, in particular in [H1]. Let 2 2 ( V ) be the algebra of polynomial coefficient differential operators on 2(V). Multiplication in 2(V) embeds 2(V) in 29(V). Let 2(V) be the subalgebra of 2 2 ( V ) consisting of the constant coefficient differential operators. There are canonical isomorphisms between graded algebras and GL(V)-modules. (3.1 a)

2(V) ~ St(V) ~ ~( V*).

(The action of GL(V) on 2(V) is via conjugation inside End(~(V)). Here St(V) is the symmetric algebra on V. Let 2t(V), St~(V), etc. denote the subspace of degree 1 homogeneous elements in 2(V), St(V), etc. For each l>O, we have canonical identifications of GL(V) modules

(3.1b)

2'(V)-stl(V)~(V*)--(2'(V))*.

Multiplication in # 2 ( V ) defines a linear map

(3.2)

~:2(v)| U(p| (p~2(V),L~2(V))

In fact the map # is a linear isomorphism of GL(V) modules. Furthermore, it exhibits a GL(V)-invariant bi-filtered structure on ~2(V). Set (3.3)

22k't(v)=l~(o~k,~1~(V)|

Then the 2~ka(V) defines a (Z + • Z+)-filtration on 22(V), that is

22~,,(v). 221.J(v)c2~*+,.,+J(v). Form the associated graded algebra [H1]. Set (3.4a)

G r 2 2 k , 1(V) = 2 2 k"l(V)/(2~k - x, 1(V) + 2 ~ k"~- 1(V)),

and (3.4b)

Gr~2(V)=

Z

k,l>O

Gr2~k'l(V) 9

The Capeili identity

573

We give GrY~(V) a structure of(Z x Z)-graded algebra in the standard way [H1]. Then the isomorphisms

fi :~k( V) | ~l( V).~ Gr~

k' l(V)

fit together to define an algebra (and GL(V)-module) isomorphism (3.5)

~ Gr~(V).

fi: ~ ( V ) |

4. Now let G~_GL(V) be a reductive subgroup. (For the moment, we do not insist that G be multiplicity free.) Consider the algebra ~ ( V ) G of G-invariant differential operators on ~(V). Since the map fi of (3.5) is an isomorphism of GL(V)modules, it gives us a strategy for constructing a basis for ~ ( V ) ~. The strategy is based on the following simple principle. Consider a group G and two irreducible representations a~, a2 for G. Then the tensor product a~| 2 will contain a G-invariant vector if and only if a 2 is isomorphic to the contragredient of a 1, which we write a2_-_cr*. If this does hold, then at| 2 will contain exactly one G-invariant, which we construct as follows. Let {y~} be any basis for the vector space on which a I realized, and let {y*} be the dual basis for a~. Then the G-invariant in al| 2 is (4.1)

Jo,o: = Z YiOY* i

Now suppose 1,'1 and V2 are two G-modules. Suppose that we know decompositions (4.2) into irreducible subspaces. In these sums, a (respectively r) is an arbitrary index labeling the summands, and a~ (respectively ap) indicates the isomorphism type of Y~= (respectively Zoo). (4.3) Lemma. Notations as above. The space ~(VI@V2) ~ of G-invariant polynomials on 111~ V2 has a basis consisting of the elements J~:o~where a, fl runs over all pairs such that ap ~-a*.

Proof. We have the natural isomorphism ~(VI~V2)~-~(V1)| decompositions (4.2) and looking for G-invariants, we find

~(v~+v2) ~ - E ( ~ . |

Using the

~

The lemma follows from this decomposition and the discussion leading to (4.1). We note two special cases of this result. (4.4) Corollary. I f :(V1) and ~'(V~') ~-5P(V2)have no isomorphism-type of G-module in common, other than the trivial representation, then ~(Vl + vd~

~(v1)~ |

~.

(4.5) Corollary. I f V1 is a trivial G-module, then ~'( vl + v9 ~ ~- ~ ( vl) | ~'( v9 ~ .

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R. Howe and T. Umeda

By specializing the above discussion to the situation when I"1= V and Vz = V*, and using the natural isomorphisms

~(v)~-~e(v)~(v*), we arrive at the following decomposition of G-invariant differential operators. (4.6) Proposition. The operators #(J~,~p) (notation as in (3.5), (4.1) and (4.2)) for all pairs ~, fl such that tr~ = op, form a basis for ~ ( V ) ~. In this proposition we implicitly use the description ~,~(v) ~ = ~(~(v)|

~.

As a subalgebra of ~ ( V ) , the algebra # ~ ( V ) ~ is filtered, and the analog of (3.5) holds: ~:(~(v)

| ~(v))G~ Gr(~(v)~).

5. The passage from ~ ( V ) a to G r ( # ~ ( V ) ~) results in some simplification of structure. In particular, G r ( ~ ( V ) a) is always commutative, whether or not ~ 2 ( V ) a is. There is a further filtration which we can impose on ( ~ ( V ) | ~ ~ - G r ( ~ ( V ) ~ ) , and which yields an associated graded algebra of still simpler structure. Let B_~G be a Borel subgroup, and let N =cB be the unipotent radical of B. Let B be another Borel subgroup, in general position with respect to B, so that (5.1)

BnB=A

is a Cartan subgroup of G. Let N be the unipotent radical of B. Then (5.2)

B= AN ,

B= AN. .

Let fl, b, u, a, ~, and ~ be the Lie algebras of G, B, N, A, B and N, respectively. We have decompositions (5.3)

0=u~a~ft,

b=a~u,

i~=a(gfi.

See [Hu] for these facts. Let ~a(V)N be the algebra of N-invariant polynomials. Since A normalizes N, it will leave ~(V) N invariant, and under the action of A, ~(V) N will decompose into eigenspaces for A. These will be the same as eigenspaces for B, since A _~BIN is the commutator quotient of B. (We can use this fact to identify a character of A with a character of B, whenever this is convenient.) Let ~(V) n' r denote the A-eigenspace in ~ ( V ) x corresponding to the eigencharacter ~p. We have the direct sum decomposition (5.4a)

~(V) s =

~. #(V) B'~. ~EA

Here ,~ is the lattice of(rational) characters of A. In fact, since the ~pwhich occur will all be highest weights of representations of G, it suffices to let tp vary in the cone + of dominant weights with respect to the ordering on weights corresponding to B. We can further refine this decomposition by breaking ~(V) up into its homogeneous pieces. This gives (5.4b)

~ ( V) s ~- ~ a #'( V)~' v "

The Capelli identity

575

Since the ~ ( V ) are just the eigenspaces for the action of the scalar matrices on ~(V), the decomposition (5.4b) will already be implicit in (5.4a) if G contains the scalar matrices. The theory of the highest weight says that if Q e ~a(v)n' ~ then the transforms of Q by G span an irreducible G-invariant subspace YQ~_t~t(V).Further, YQc~t~Z(V)s = CQ. Hence if we have a decomposition (4.2) for V = V1, then any choice of a nonzero element in Y~n~(V) N gives us a basis for ~Z(V)N compatible with the decomposition (5.4b); and conversely, any such basis {Q~) gives us a decomposition like (4.2). Consider an irreducible module Y for G. The theory of the highest weight implies we have a decomposition

y= yN~fi(y),

(5.5a)

where yN is the line of highest weight vectors and fi(Y) is the span of the images x(Y), xefi. Consider the dual space Y* of Y. It has a parallel decomposition Y*= Y*S@n(Y*).

(5.5b)

Note we have reversed the roles ofn and fi in going from (5.5a) to b). Since Y~ is the space of vectors annihilated by all x ~ n, one can check from the definition of contragredient representation that the two decompositions (5.5a) and b) are dual to each other. That is (5.6)

r*~ = fi(Y)•

rt(Y*)= (yS)•

where l means the annihilator in the dual. It follows that if {y~}is a basis for Y. with Yl 6 YN and yj ~ ft(Y) for j _> 2 then for the dual basis {Yi, }, we have Yl, E Y ,f~ and yj E n(Y*) for j > 2. Hence the G-invariant element

~i y,|174

(5.7)

-~- i~>=2 yi|

belongs to (yN| y,a)~(ft(y)| From this paragraph, we conclude we have a decomposition (5.8a)

~(V) = ~(V)S(9 ft(~(113).

Dual to this is another decomposition (5.8b)

~(V) = ~(V) ~(~n(~(V)).

Just as for ~(V) N, the space ~(V) a will be invariant under A. Indeed, by the discussion of the previous paragraph, the space ~t(V)S is dual to ~*(V) N, and the actions of A on the two spaces are mutually contragredient. Thus we can write (5.9)

~'(V) s = Y. 2'(V) ~' ~-',

where the lp's involved are the same as those in (5.4b). More precisely we have ~ ( v ) B. ~ -' u (~"(v)', ,~)*.

(5.10)

In particular, these spaces have equal dimension. Taking the tensor product of the decompositions (5.8a) and (5.8b) gives a decomposition (5.11)

~(V)|

~--(~(V)N| + ~(v)|

(ri (~(V))|

576

R. Howe and T. Umeda

Since #(V) ~rand ~(V) ~ are both A-modules, their tensor product has the structure of an (A x A)-module. Let An_gA x A denote the diagonal subgroup. Let zoo denote the projection of ~(V)| onto #~(v)N| N associated to the decomposition (5.11). (5.12) Proposition. The projection 7ro maps (~(V)|

(~(V)N|

G isomorphically onto

A~, the space of Ad-invariants in ~(V)N|

~.

Proof. Indeed, each summand Y,.| Yo*~in the decomposition (4.2) of~k(v)| intersects #k(V)N| in a line, ~nd this line will be the image of Y, | Y,~ under the projection rro. The A-eigencharacter on the hne ~ . characterizes the isomorphism class of representation Y,., by the highest weight theory; likewise the A-eigencharacter of Y~ characterizes Y,~. Two A-eigenlines define mutually contragredlent representations if and only if their e~gencharacters are mutually inverse, if and only if their tensor product defines the trivial representation of An. We have seen, in the discussion leading to (5.7), that for an irreducible G-module Y, the eigenlines yN and Y*s are mutually contragredient. Since these lines determine the isomorphism types of their G-modules, it follows that the lines Y~ and Yo~ define mutually inverse A-eigencharacters if and only if Y~. and Y, are equivalent. The proposition follows from these remarks, Proposition 4.6, a~d (5.7). .

.

.

.

/~r

,

c*

r

6. The spaces (~(V)| ~ and (~(V)N| a~ are both subalgebras of #(V)| The isomorphism no between these spaces is not an algebra homomorphism. However, it can be modified to become an algebra isomorphism. Let us abbreviate (6.1)

(~(V) u | ~(V)~) a~ = ~r

We note that ~r is still invariant under A • A. Since A n acts trivially on ~r we effectively have an action of(A • A)/Aa on ~r It will be convenient to identify this quotient with the first factor in A • A. Thus if Lx c=~(V) N and Lz _-_~(V) ~ are two mutually contragredient A-eigenlines, we consider the character of (A x A)/Aa acting on La| to be the same as the character of A acting on L~. This action of A on ~r defines a grading on ~r by t] +, the semigroup of dominant characters of ~ . That is, if ~r is the ~p-eigenspace for A, we have ~r176162176 ~ d ~ , ~: (Wi~$+). Combining the ~ +-grading with the usual degree, we obtain an ($ + • Z +)-grading on ~r Abbreviate (6.2)

(~(V)|

= ~.

The algebra ~ is not graded by .~+, but it has a natural/]+-filtration. Given two characters Wa, W~ of A, we will say ~ ~< ~ z if ~p~pi- ~is expressible as a product (with positive exponents) of the positive roots of A (i.e., the characters of A which occur as eigencharacters in the adjoint action in n.) We know that ~ has a basis of elements at,# as described in (4.3). Each J,a is constructed using a pair of equivalent irreducible representations of ~(V). For ~ ~ A+, let ~ ' ) denote the span of those J,# which arise from representations with highest weights less than or equal to ~, in the sense just described. (6.3) Lemma. The subspaces ~o) of ~ define an .~ +-valued filtration of ~. That is

~ ' ) ~ ' ) ~_~ ' ~

OP,q~~ ~ +).

The Capelli identity

577

Proof. Consider J~pe Y~|176 and Jr~e Yr ~Y~.,. If J ~ , e ~ , then the highest weight a~ = cr~ satisfies a~ < o2. Similarly, if J~a e ~C~,},then" % = a r < (p. The product J~Jr~ will be inside (Y~Yr174 In particular, we can expand J~aJ~ as a linear combination of terms J~,, where a, is equivalent to a G-submodule of Y,Y,, which is a quotient of the tensor product Y,.| Y~. It is well-known [Hu, p. 14~, Ex. 12] that the highest weights appearing in Y,,| are all less than or equal to a~a~; the lemma follows. Consider the graded algebra Gr~ associated to this filtration of~. Precisely, as space we have G r ~ = ~ ( ~ ' / ~. ~"P),

(6.4)

G r ~ = , ~ + Gr~ ~.

The multiplication in Gr~ is the direct sum of the bilinear Gr~ ~ • G r ~ ' ~ Gr~ ~q' maps which are quotients of the multiplication maps ~<~) x ~ ( ~ ) ~ ' ~ ' ~ . Observe that the span of the J~p (as in (4.1)) such that a, = ~pprovides a linear complement to ~ ) in ~ ) , and so provides a canonical representative for each element of &~. This observation implies that the linear isomorphism r~0 of Proposition 5.12 can be restricted and factored to define linear isomorphisms between ~ and ~r Taking the direct sum and all these gives us a linear isomorphism

Gr:zo : Gr~-~ ~z/ .

(6.5a)

By virtue of its construction, this ~o is a graded isomorphism:

Gr~to(G e ~ ) = ~r

(6.5b) for all ~pe.~ +.

(6.6) Proposition. The mapping r~o of(6.5a) defines an isomorphism of 71+-graded algebras from Gr~ to ~ .

Proof. Consider two elements J~a and J # of ~, belonging to subspaces Y,,| Y,~ and Y~| Y~, of~(V)| In forming the elements J~a and Jra according to the recipe (4.1) we have the freedom of choosing a basis for Y~. and for Y,. Let us choose our bases following the prescription of (5.7). Thus we choose a basis {Yi}, l < i <_dim Y~.such that y x e Y~ and y~e fi(Y~,)for i > 2. We can further assume that all the y~are A-eigenvectors. Their eigencharacters will then be less than a~. Choose a basis {zi}, 1 _<j_
v "z

i.j

~,,*z*-,,

z ~,

*z *•

i+j>-2

TO compute ~o(d,aJ~) with r~0 as in (6.5a) we should apply the no of Proposition 5.12, from & to ~ , and then further project to ~ . If we inspect the sum just above which equals J~Jr~, we see of all the summands y~zy| except for y~z~| belong to A-eigenspaces for characters of A less than crierr. We conclude that f~o(J~J~)=y~z~| proving the proposition. 7. The considerations of Sects. 4--6 become simpler and sharper when applied to a G which acts in a multiplicity-free fashion. If G is multiplicity-free, then the decomposition (4.2) reduces to the completely canonical decomposition (2.3). Since there is at most one summand of a given isomorphism type, we can dispense with the auxiliary parameter ~, and label each summand by its highest Weight. This

578

R. Howe and T. Umeda

just gives (2.3). Since the summands Y~ are canonical, the basis elements J~v, of (#(V)| a give a canonical basis for this space, and similarly their images #(J~) give a canonical basis for ~'~(V) a, and their images no(J~) give a canonical basis for the algebra a'=(#'(V)N| A'. Further, all the homogeneous components are one-dimensional. Also, alone for multiplicity-free actions, the algebra #~ is commutative. (7.1) Proposition. The algebra ~ ( on ~(V) is multiplicity-free.

V) ~ is commutative if and only if the action of G

Proof. If the action of G on ~(V) is multiplicity-free, then any endomorphism T of ~'(V) which commutes with G must preserve each summand in the decomposition (2.3), by Schur's Lemma. Further, the restriction of T to a given summand must be scalar, again by Schur's Lemma. Hence g ~ ( V ) ~ is a subalgebra of the multiplication algebra on the set {Y~o}of irreducible summands of ~(V), and so is abelian. On the other hand, suppose ~(V) is not multiplicity-free. Thanks to the "unitarian trick" we know there is a projection map R: ~'~( V)--, ~ ~( V)~ (the "Reynolds operator" [MF, pp. 26-27]) with the property that i) If a ~ ~@(V), b ~ ~ ( V ) ~ then

R(ab) = R(a)b ,

R(ba) = bR(a)

(i.e., R is a ~(V)G-bimodule map). ii) If X=~(V) is a G-invariant subspace, and a ~ ( V ) and

alxEEnd(X) a,

then

is such that a(X)C=X

g(a)lx=alx.

It follows from these properties, and the fact that ~ ( V ) is strongly dense in End(~(V)) (i.e., given finite dimensional X~_~(V), one has ~ ( V ) l x =Hom(X,~(V))), that ~ ( V ) ~ acts irreducibly on ~(V) ~'~ for any ~pe,,~+. Hence ~ ( 1 / ) ~ cannot be commutative unless dim~(V) n'~'= 1 for all ~p, i.e., G is multiplicity-free. 8. In addition to the relations between 9a~(V) ~, (~(V)| ~, and (~(V)~| a~ detailed above, we have a natural isomorphism (cf. (3.1a))

(8.1)

(~'(v)|174174

~ ~(V@V*) ~.

When G acts prehomogeneously, especially when G acts multiplicity-free, this isomorphism provides another interesting incarnation for ~ ( V ) ~. Suppose G acts prehomogeneously on V*. Fix an element 20 in the open G-orbit in V*, and denote the stabilizer in G of ,to by H. The affine subspace V + 2o__gV ~ V* is stabilized by H, and the action of H on V + ;to is equivalent, via translation by -2o, to the linear action of H on V The G-invariant set in V ~ V * swept out by V+2o is G(V+20)= V+G(2o), which is dense in V(~ V*. Therefore, any G-invariant polynomial on V ~ V* will be determined by its restriction to V+2o. Furthermore, this restriction must be H-invariant. Translating by - ~ o gives us an H-invariant polynomial on V Thus

The Capr

identity

579

we obtain an injective homomorphism (8.2)

ro : H( V@ V*)~--. H( V)n; (roQ)(v)=Q(v,,~o}

(v~ lO.

The situation is especially sharp when the G-action is multiplicity-free. (8.3) Proposition. a) Suppose G acts prehomogeneously on V*. Let H be the stabilizer in G of an element 20 in the open G-orbit in V*. There is an embedding r o : H(V@ V*)G--,H(V) n described in (8.2). b) I f the action of G on (on V or V* ) is multiplicity-free, then the embedding ro is an algebra isomorphism. Remarks. a) Part a) of the proposition is valid whether or not G is reductive: it requires only that G acts prehomogeneously on V*. If G is reductive, then one can show IS, Proposition 14] (see also [SK, p. 71]) that G acts prehomogeneously on V if and only if it acts prehomogeneously on V* (see also [BW, p. 42] for the essential fact in the argument). b) Since Hs(V *) ~ HI(V)*, the action of G on V will be multiplicity-free if and only if the action of G on V* is multiplicity-free. In particular, if H* ___G is the stabilizer of a point Vo in the open G-orbit in V, we have

(8.4)

H(v) R H(V

V*)

H(V*) n" .

Proof Part a) is already proved. Consider part b). We have the decomposition (2.3) of H(V). In formula (4.1) (see also the first paragraph of Sect. 7) we constructed a basis {J~w} for (H(V)| ~. The element J ~ of H ( V | V*) corresponding to J ~ via the isomorphism (8.1) will be

where {Yi} is again a basis for Y~o, and the 37* are the dual basis for the contragredient module (Y~)* _c__H(V*). From the definition of r 0 we have

ro(7 0 (v) = 2. In particular, ro(Jrw)E Y~o.Therefore the image ro(H(V@ V*) ~) inside H(V) n has a basis consisting of the elements ro0'. ~), one inside each summand Yr" Hence if we can show that dim(H(V)nn Y~,)_<1 itor all o2, the proposition will follow. Since the action of G on V* is multiplicity-free, a Borel subgroup B of G will have an open orbit on V*. This means that B has an open dense orbit in G(2o) ~- G/H, which is the same as to say, there is an open dense (H, B) double coset in G. The theory of the highest weight says that every irreducible G-module can be realized on a space of functions on G which transform under right multiplications by B by some character of B. The action of G on this space is just by left translations. Thus a vector which is H-invariant will correspond to a function invariant under left multiplication by H on G. Clearly this function is uniquely defined up to multiples on any (H, B) double coset. In particular, there is at most one such function on the open dense double coset. By continuity there is at most one such function on all of G. Hence, for any irreducible G-module Y, one has

580

R. Howe and T. Umeda

dim(Yn)
ro

~(V*)R'~-#(V ~ V*)~-~(V) H. The two algebras ~ ( V * )H* and ~(V) H have ~ + (V)-filtrations defined on them, and if ~ ( V ~ V*) ~ is given the ~+(V)-filtration transferred from (~(V)| ~ then ro and r~ are filtration-preserving. viii) The algebra ~(V) ~ is a polynomial algebra on ~(P~+(V)) generators. It allows A as a group of automorphisms. It has a generating set in canonical bijection with P g +( V), each element of which is canonically defined up to multiples; the set as a

The CapeUiidentity

581

whole is uniquely defined up to the action of A. In particular, each of the algebras

~(v) ~ (~(v)| ~ ( v ~ v*) ~ ~ ( v ) '~ ~,(v) N

~ (~(v)~| ~(v*)'"

~

is a polynomial algebra on ~(P$+(V)) generators; hence all these algebras are mutually isomorphic. Remarks. a) Of the seven isomorphic algebras listed above, the ones which are usually the easiest to compute directly are either one of ~(V) H or ~(V*) n*, and ~(V) N. Theorem 9.1 describes how to ascend from knowledge of one of these to ~ ( v ) ~. b) The algebras ~ ( V ~ V*) a provide interesting examples of actions in which the invariants are a polynomial ring. Such actions are sometimes called co-regular [Schl]. The co-regular actions of simple groups have been classified in [KPV]. 10. Having studied ~ ( V ) ~ as a thing-in-itself, we turn to investigation of its relation to G, or more precisely, to ~(g), the universal enveloping algebra of the Lie algebra g of O. It will be convenient now to write the action of G on V as a representation. (10.1a)

0: G~GL(V).

By differentiation of one parameter groups at the origin we derive from Q a homomorphism of Lie algebras (10.1b)

q: g-~gl(V)_~ ~ ( V )

which extends to a homomorphism of associative algebras

(10.1c) Let ~ ( g ) (10.2)

q: ~ ( ~ ) - ~ ( v ) . denote the center of q/(g). Then clearly e ( ~ ( g ) ) _-r~ 2 ( V ) ~ .

From our example in Sect. 2, it is natural to ask (10.3) (The abstract CapeUi problem). When is the inclusion (10.2) an equality? When is 0: ~ ( g ) - ~ ' ~ ( v )

G

surjective? If inclusion (10.2) is an equality, then one can pose: (10.4) (The concrete Capelli problem). Express the canonical generators of ~ ( V ) ~ as images by 0 of elements in .~//(g). Comments on this statement are in order. First, if Q is not faithful on ~r then the answer to (10.4) will be ambiguous. Hence a complete answer to (10.4) would include a description of Ker(Q)n~q/(g), and perhaps also a description representative of the canonical generators of ~ ( V ) a which are in some sense "bast". Second, one could be even more ambitious than (10.4), and ask to express the elements of the whole canonical basis for ~ ( V ) ~ in terms of ~q/(g). This would be accomplished by solving (10.4) and then expressing the canonical basis

582

R. Howe and T. Umeda

elements of #~(V) a as polynomials in the fundamental generators. This problem, which is of independent interest, has been considered in [KS] for hermitian symmetric domains of tube type. Third, in the other direction, again referring to our example of Sect. 2 explaining the classical Capelli identity, the most interesting case of (10.4) occurs when there is a G-eigenvector Qo-~~(V). Then there will be a contragredient G-eigenvector Lo ~ ~(V), and the product QoLo

will be in ~ ( V ) a. It is already quite interesting to express this as an element of ~,/(g). In the example of Sect. 1, this is the Capelli identity. In the following section, we study problems (10.3) and (10.4) for Kac's list IK] of irreducible multiplicity-free actions. We answer question (10.3) for the whole list, and we answer (10.4) for a large part of it. Before proceeding to the specific calculations, however, we make the connection with the Double Commutant Theorem. For a subalgebra d _~~ ( V ) , recall that d'={a'e~(V):a'a=aa',

all ae~r

is the commutant of~r and d " = (d')', the double commutant o f ~ . It is obvious that ~r ~r and that d " = ~". (10.5) Proposition. Let q: G ~ G L ( V ) be a multiplicity-free action of the connected reductive group G on E Then ~ ( V ) ~ = 0(*Z(~))'= 0(~ZqZ(g))". Hence #~(V)6=q(~eq/(g)) /f and only if Q(.Zq/(fl)) is equal to its own double commutant. In particular, if Q(ad(9)) is its own double commutant, then ~ ( V ) ~ = e(~r0a(g)). Proof. Since G is connected ~ ( V ) (~ is the same as the set of operators which commute with Q(g),which in turn is just Q(ql(g))'.Thus the first equality is evident. To show ~(V)~=Q(2Zq/(g)) ", observe that, since Q(q/(fl))__q(s we certainly have Q(-~ead(g))"~ # ~ ( V ) ~. On the other hand, consider the decomposition (2.3) of~(V) into irreducible G modules. The elements of ~q/(9) act on each irreducible summand Yv by scalars: e(z) (y)= X~(z)y ,

where X~ is the infintesimal character [Hu, p. 129] of Y~. A basic fact [Hu, Sect. 23.3] of the representation theory (essentially the Harish-Chandra homomorphism) says that X~ determines ~p: a finite dimensional representation is determined by its infinitesimal character. It follows that any L e q ( ~ ( ~ ) ) ' must preserve each Y~. Hence, @~(V) ~, which acts by scalars on each Y~, will commute with L. The proposition follows.

The examples 11. In this section, we treat the Capelli problems (10.3) and (10.4) for the irreducible multiplicity-free actions, classified by Kac in [K]. Specifically, we

The Capelli identity

583

consider the following linear group actions.

11.0.1)

i) GL.| ii) S2GLn iii) A2GL,, iv) O.|

vi) Sp2.| vii) Sp2.| viii) Sp,,|

1

3

ix) Spinv|

xi) Spin9| l xii) G2| 1 xiii) E6|

1

1

v) Spzn| l x) Spinlo| 1 In this list, the notation is intended to suggest the action. Thus GL,| signifies the action of the group GL, x GL,, on the vector space C " | and similarly for the Spx GL pairs. The Spin groups act by their basic spin representation. The symbols AZGL, and SZGL, indicate the action of GL, on the skew symmetric and symmetric 2-tensors respectively. The action of Gz is on its 7-dimensional module, and the action of E6 is on its 27-dimensional module. The factor GL 1 occurring in many of the entries indicates the scalar matrices on whatever vector space supports the action. This list appears to differ slightly from Kac's list. The main reason is that all our groups include the scalar matrices, while some on Kac's list do not. This is because one can easily see that without the scalar matrices, there is no way to obtain the Euler operator as an element of Q(q/(g)), so that abstract Capelli problem will trivially have a negative answer. (11.0.2) Lemma. Suppose the homomorphism Qof (10.1)maps G to SL( V). Then the Euler operator, which takes the form

Z xi i

OX i

in any system of coordinates on V, does not belong to Q(q/(~)). Proof. There are several ways to see this. One is to note that, under the natural action of gl(V) on the distributions 5P*(V), the Haar measure dv is annihilated by every element of ~I(V). Hence if e(g)~ ~I(V), Haar measure will be annihilated by every element of Q(~q/(fl)). But the Euler operator does not annihilate Haar measure. Thus if the Euler operator is in #(q/(g)), it is in e(q/(g))- Q(gq/(g)). But an element of Q(q/(g))-e(gq/(fl)) will not annihilate the constant function, while the Euler operator does annihilate the constants. Remarks. a) From the form (4.1) of the operator J~o~0,the basis element of ~ ( V ) ~ corresponding to ~pe / ] +(V), we can see that it is characterized up to multiples as an element of ~ ( V ) a by two properties: (11.0.3a) J~,~,has order (as a differential operator) (less than or) equal to the degree of Q~0. (11.0.3b) J~,~,(Qw,)= 0 for ~p4: ~p', deg Q~, < deg Q~,. This characterization of J ~ will figure importantly in our computations. In fact, we will see the J~0~,tend to have much more extensive vanishing properties than the characterizing property (11.0.3b). b) Although we have defined, in (4.1) and Theorem 9.1 a canonical set of generators for 9 ~ ( V ) a, in the calculations which follow we will be satisfied to compute them up to scalar multiples, as suggested by the characterization (11.0.3),

584

R. Howe and T, Umeda

and we will be cavalier about whether we have the correct normalization of these operators. We will now treat individually the actions of list (11.0.1). The results of our discussion is summarized in Table (15.1). Conversely, this section may be regarded as the justification for the table. 11.1. Case i. GL,| This is the action on which Capelli worked [Cal; Ca2; Ca3]. The special case when n = m was discussed in Sect. 1, and the notation used there will, mutatis mutandis, be used here. In particular, the space on which GL. x GL~, acts will be M,m, the n • m matrices, and the coordinates of T e M , m will be the usual matrix entries (t o, 1 ~ i ~ n, 1 < j < m}. We have a (GL~ • GLm)module decomposition (11.1.1)

~(M,m) ~- Y.q,|176 o D

analogous to (1.13), but where the Young diagram D is regarded as an infinite sequence of decreasing non-negative integers al>a2>-->_...>ak>ak+~ > .... of which at most min(n, m) are non-zero. The largest index I such that at 4:0 is called the depth of D. The generators of A+(M,m) are the weights D k x D k where (11.1.2)

Dk=(1, 1.... ,1,0, 0, ...)

1
consists of k l's followed by zeroes. The corresponding representations of GL, x G L , are (11.1.3)

O~174

~ A~(C")| A~(C') .

The corresponding fundamental B-eigenvectors are the determinants ?k described in formula (1.15). (We take the product of standard upper triangular Borel subgroups of Gin and GLm for our B.) By the recipe (4.3), we can construct the canonical generators for the algebra of differential operators as follows. Let I = { i l , i 2..... ik}_~N= (1,2, 3, ..., n} d={j~,J2 .... , A } ~ M = ( 1 , 2 , 3..... m } ~

i,
be subsets, both of cardinality k, of N and M respectively. Let

I t~lj1 ... thj k ] (11.1.4)

T.=

"

"

Ltiujl

~ildkJ

be the submatrix of T whose rows are selected from I and whose columns are selected from J. Let (11.1.5)

C~(T),j =

"

"

LCnt~d~.'" (~i~jk be the dual matrix gotten by replacing t,b with d,b = d--~b"The kth generator of ~ 2 ( M ~ ) ~215 ignore) (11.1.6)

can then be written (up to a normalization factor which we Fk= ~ det(TQdetd(T)u.

l,J

The Capelli identity

585

The same reasoning as given in Sect. 1 (see (1.8)) tells us that the abstract Capelli problem has a positive answer in this case. In fact, something more precise holds. To state it, let the Eo~I < i, j < n) be the standard basis of the Lie algebra gin, and let E~(1 < i,j < m) be the standard basis of glm. Define matrices

[

Ei~i:+k-1

(11.1.7)

1"Ii=

Ei2il.

E~: Ei~i:~. k -

... Eilik] 2

Eiki2

Eikh

Ei2ikll"

Ei~i

Let det Ht be the element of q/(gln) obtained by taking the usual alternating sum of products of entries of lit, taking care to do the multiplication in the products in the order indicated by the rows. Define matrices /-/~, with entries E~d~, and det/-/~ ~ ~(glm)in parallel fashion. Finally, set (I 1.1.8)

C k = ~.. det//, I

and define C~ e q/(gl~ analogously. We call

qz@3.

Ck (or

C~) the k-th

Capelli element of

Let L denote the action of GLn or aS(gin) on ~(Mm), and R the action of GLm or q/(glm). Here L is taken in the modified form (1.4'), and R is as in (1.3) (except in the formula for R(E~j),the sum is from 1 to m). Then the k-th Capelli identity I-Ca2; Ca3; H1] says

(11.1.9)

L ( f k ) = FR = g(c'k).

A conceptual proof of this identity was given in EH1]. Because this argument transfers well to cases ii) and iii) of symmetric and antisymmetric matrices, we review here the main pieces of the argument of [H1]. (We treat only Ck; the arguments for C~ are the same.) (11.1.10) ai) The operator Fk is of degree k as a differential operator. aii) Fk annihilates all subspaces Om@Qm o o with depth (D)< k. aiii) Properties ai) and aii) characterize F~ up to multiples as an element of bi) The operator L(Ck) has properties ai) and aii) also. bii) The element Ck of ~(gln) is in fact in . ~ ( g l , ) , so that L(Ck) is in

~(M~m)GL~ • ~L,,

C) The operators Jk and L(Ck) are equal on the level of symbols, their top order terms agree. Formula (11.1.9) clearly follows from these facts. Most of the facts are themselves easy to check. The degrees of Jk and L(Ck) are obviously both k, and it is an easy calculation (based on identities resulting from the fact that the det T~are the matrix entries for the action of GL~ on Ak(C~) - classically known as the BinetCauchy identities [T1] to check that their top order terms agree. Fact aii) can be checked easily by applying Fk to highest weight vectors, which are explicitly known (1.16). The same calculation establishes the analog of property aii) for L(Ck), providing we know bii), which guarantees that L(Ck) acts by a scalar on each D Q~ | D (These essence of this argument and calculation was given in establishing (1.23).) Both facts aiii) and bii) follow from this calculation, since one sees from it that the algebra generated by R(C~), 1 <j
586

R. Howe and T. Umeda

Thus we are left with needing to show that the Capelli elements Ck belong to the center of J(gIn). An argument for this was also given in [H1], but it was not particularly instructive (and in fact was slightly wrong). We give here another argument which we feel better reveals what is going on. Consider the q/(gln)-valued polynomials (11.1.11)

Ck(z) = ~ det(H1 -- z). i

Here again I is a subset of N of cardinality k, and/-/, is the matrix defined in formula (11.1.7). For any function f of z, consider the difference operator

Af(z) = f(z + 1) - f(z).

(11.1.12)

If A is any n x n matrix, then we have 3 (det A - z) = det(A - z -- 1) - det(A - z) = -- ~ (det(A - z - P j_ 1) - det(A - z -- P~)), j=l

where P~ is the diagonal matrix with its first j diagonal entries equal to one, and other entries equal to zero. Expanding these determinants along thej th column, we find that det(A - z - P j_ l) - det(A - z - P j) is equal to the determinant of the (n - 1) x ( n - 1 ) submatrix of A - z - P j obtained by deleting the fh row and column. Applying this computation to Ck(Z) gives us the formula (11.1.13)

A Ck(Z) = - - (n - - k + 1 ) C k _ 1(z).

Formula (11.1.13) implies that, to show Ck is ~q/(fll,), it suffices to show that For this, see the Appendix.

C,(z) is in s

Remark. A corollary of the above arguments is that the Capelli elements Ck, 1 --
(11.1.15)

Q (Ck)= E Vi (a,,+k-O. * /=1

Remark. We note that this number is independent ofn > k. For k = n, this is just the product fi (a i + n - i ) . Slightly more generally [----t

(11.1.16)

q~(C,(z)) = f i (a, + n - - i - z). i=1

It follows from this and (11.1.13) that the algebra generated by the 0,~ for 1 ~_j
The Capelli identity

587

Chandra homomorphism [Hu]. We will call any polynomial in the Ck with constant (i,e., independent of n) coefficients a stable element of ~rq/(91.), 11.2. Case ii. S2GL.: This case was considered by Gfirding [G] and Turnbull IT2]. The space on which the action takes place is $2(C"), which we will think of as the space of symmetric matrices. We denote a typical element of the space by T, with entries tij (1 < i, j < n) with the understanding that tij = tji. We denote the action of GL., and of q/(gl.) on ~($2(C")), by a. The standard basis elements Eij of gl. are represented by the linear vector fields (11.2.1)

a(E 0

-~ ~, (tiadja + taiday ) . a=l

(In applying this formula t o and tji should be treated as independent variables.) Again we use the notation of (1.4) for partial differentiation. It is well-known (see [H1; Shl; Th] for example) that the decomposition of ~i~(S2(Cn)) under the action of gl. is described by

~(s2(c")) ~ (11.2.2)

E Qno , ~ .... depth (D) ~ n

where the condition that D = (a 1, a2 .... ) be even means that its entries a~ are even integers. The fundamental generators of ~ +($2(C")) are the weights 2Dk, with Dk from (11.1.2). The corresponding highest weight vectors are (11.2.3)

tlk= det

"

" .

Lt~1 ... t~kj

We can construct the canonical generators of ~ ( $ 2 ( C " ) ) ~L, by recipe (4.3). In this regard, we note that we can write the Euler degree operator F1S= 1
in the more symmetrical form F1S : ~

+ ~ij)tijOij. i=1 j=

Here the 1 + 6o is necessary to compensate for the fact that when i , j , the term hjaij -~-tjiOji g e t s counted twice in the sum. This expression may also be written Fs = trace

T~(T),

where

(I 1.2.4)

~r(T)=

20xt

012 ... 01.]

021

2022 ... 0 2 . .

9

i

"'.

"

We can express the higher order invariant differential operators in similar fashion9 Precisely, we have (11.2.5)

Fff = E det(T1s)det(~(T),s). l,J

R. Howe and T. Umeda

588

The notation hem is similar to case i). Both I and J are subsets o f N ofcardinality k, and Txz denotes the submatrix of T obtained by eliminating the rows not in I and the columns not in J. We should note that (11.2.5) is not really an instance of (4.1), because the det(Tij) are not linearly independent. Nevertheless, it may be verified that Fks is invariant under a(GLn). For most of the argument which follows, it is not important what the exact form of F~s is. The main fact needed is simply that the dct(T~j) span a ~z(GLn)-invariant subspace of #~k(s2(cn)). This subspacc will necessarily contain the k-th basic representation, viz. 0~~'. More precisely, we need the dual fact about the det(~(T)1z). This will allow us to characterize the Fks. (11.2.6) Proposition. a) ['ks is of degree k as a differential operator. b) Fks annihilates 0 ff whenever D has fewer than k rows. c) Fks is characterized up to multiples by facts a) and b). d) (Capelli identity for S2GLn): We have (11.2.7)

FkS=a(Ck)

1 <_k
where the Ck are the Capelli elements for s

constructed for case i (11.9).

Remark. The case k = n is proven by Turnbull IT2]. Proof. Statement a) is obvious. For statement b), it is enough to check that Fks annihilates the highest weight vector of On~ of depth (D)< k. The fundamental highest weights are given by (11.2.3). From that formula, we see that if depth (D) < k, then the highest weight of 0n~ depends only on the variables tij with i, j < k. But from formula (11.2.3), or from the weaker fact noted below it, we see that every term of Fks involves differentiating with respect to a variable tij with i or j greater than or equal to k. Hence Fks will annihilate the highest weight of On~when depth (D) < k. Thus statement b) is proved. We prove statement c) by induction on k. From our characterization of C k as an element of s (see Remark (11.1.19), we see that if c) holds, then d) is true up to multiples. Thus if statement c) is true for k < ko, statement d) must be also true (up to a scalar factor) for k < k o. But the algebra generated by Ck, k < k0, acts faithfully on the On~ with depth (D)
2(E~j)= ~ (h~

t~

~

As in Case ii, to apply this correctly, we should regard the t , as independent.

The Capelli identity

589

It is well-known [H1; Shl; He2], etc.) that the decomposition of ~(A2(Cn)) under the action of ~I~ is described by

(11.3.2)

~(s~)c~ ~ Y e~. D

where D runs over Young diagrams with columns of even length, i.e., D has the form

O=(bi,bDb2,b2 .... ). In other words, if we use the usual notation D = (a~, a 2, a 3.... ), then we have a2~- 1 =a2i=bi. The fundamental generators of.4 +(A2(C")) are the D2t, with Dk as in (11.1.2). The corresponding fundamental highest weight vectors are expressed in terms of Pfaffians [W; J; Shl]. These can be writtea

CPk=

Y.

sgn(J)tz~l)Jt2)tjo)J~4r..tj(2k- 1)d(2k)"

(11.3.3)

J r S2k]Bk

Here B k is the subgroup of the symmetric group S2k consisting of elements which preserve the collection of pairs {{1, 2}, {3, 4},..., { 2 k - 1 , 2k)}. The first three ~0's are tPl = t 1 2

~2=t12t34--t13t24+t14t23

r + t14t23t56 - - t14t25t36 + t t 4t26t3s -- t I 5t23t46 + t14t24t36 -- t 15 t 26t34. + t16t23t,5 -- t16t24t35 + t t 6 t 2 5 t 3 4 .

We can obtain a basis for Qff~-~A2k(C") by the following procedure. Let I = {il, i2.... , i2k) be a subset of N = {1, 2 .... , n} ofcardinality 2k. Let Pf~ denote the Pfaffian Ck formed as in formula (11.3.3), but using the xioib with ia, ib from I rather than from {1, 2 .... ,2k}. Then the Pf~ form a basis for 0ff~. We can form a dual basis by using the ~ioib rather than the xio~ to form the
FkA= V~k (Pf~)OPf~.

Because of the form (11.3.3) of the Pfaffians, i.e., because Pf~ with Ill = k is a sum of products, each of which involves at least one variable which does not appear in
590

R. H o w e a n d T. U m e d a

We will deal with the Cka in terms of their action on representations 0,D. Guided by the form (11.1.15) of the values of the C k on QD, with D=(at, a2,a 3, ...), we propose that polynomials C~(z), analogous to the polynomials Ck(z) of formula (11.1.11) should be defined by k

(11.3.6)

Q,D(Ca(z))= Y, l - I ( b i , + 2 ( k - / ) - z )

for

D = ( b , b a , b 2 , b z , b3,b3,...).

Ifl=k l = t

We will justify this definition shortly by showing how to express the Ca in terms of the Ck, and vice versa. First, concerning the relations among the Ca, we observe that, by a similar argument, we can establish an analog of (11.1.13): (11.3.7)

A 2Cg(z) = -- 2(rn - k + 1)Ca_ t(z),

where ra = [n/2] is the greatest integer less than n/2, and (11.3.8)

A2(f)(z) = f ( z + 2 ) - f ( z ) = (A + 2)A(f) (z),

where A is the standard unit increment difference operator, as in (11.1.13). Using the standard inversion formula [Bo] of the calculus of finite differences, we conclude from (11.3.7) that, for m = [n/2] we have C,,(z) a = C a, , - zC A_ 1 + z(z - 2)C~_ 2 + . . . + ( -- 1)kz(z-- 2)...(z - 2k + 2)CA_k + . ' -

(11.3.10)

m

=

E

j=O

where we have abbreviated C~(0)= Cam,and have written j-1

Q](z)= 1-I ( z - 2 c ) . c=O

Next, we observe the relation, for D as in (11.3.6): 2m

(11.3.11)

0,D(C2m(Z)) = ~I (al +2ra--i--z) i=l

= ~I ( b ; + 2 m - 2 j + l - z ) ( b ~ + 2 m - 2 j - z ) j=l

=

D

A

(C.(z -

D

A

(C.(z)).

Thus we have

C2.(z)= Ca~(z)Ca~(z-1).

(11.3.12)

Using (11.3.10), (11.3.12), and (11.1.13), we can obtain expressions for the Ck in terms of the C/t (11.3.13). Proposition. Take n = 2m. Then for any k <-m, we have Ck = ( - 1)k((n- k)!) - 1A2m- k(Ca(z)C~(z - 1))[~=o =

Z a+b~_k

2 max(a, b) _ k

.k

f~Af~A~

CaA CbA + a+b=k

. k f~Af, A t'labt...,a t.., b a+b
2 max(a, b) _ k

The Capelli identity

591

where q,,k b are rational numbers independent of m, given explicitly by the following formulas (k_2b)!2.+b_ ~ - ~ ) !

if a>b,

r/~,b= [ ( k - 2 a - 1 ) !2a+b-k

(11.3.14)

[ ( 2 - ~ - Z _ ~ Z _ ~ _ - - b ) ! if

a
Remark. The authors thank Doron Zeilberger for providing an explicit evaluation of the r/~b. We originally had only shown that the r/~b were rational and independent of m. The argument below is Zeilberger's. The proof uses a preliminary lemma. Let [z]a = z(z- 1) (z - 2)...(z - a + 1) be the descending factorial polynomial of length a. The polynomials Qa in (11.3.10) are the analogs of the [z]a, but with gap 2 rather than 1 between factors.

(11.3.15) Lemma.

I b-a 1p(b_a+s) ! [Z],+b_~ X (-- , ( b U - - a = ~ Q=(z)Qb(z- 1) = ' =o a-b- 1

if b>a,

-- 1 AV S)! [Z']a+b_ s

Z (-1) '(a-b ~=o ( a - b - 1 - s)!s! 2s

if b
Proof. This is a routine exercise using the method of [Zb]. A more conventional argument may be given by observing that the right hand side is

[z]a+b.2Fl[b-a+l,

-(b-a);

1/2]

-(z+a+b-1) where 2F1 indicates the usual Gaussian hypergeometric function [Ba]. The lemma then follows by a manipulation of the F-functions in (3), p. 11 of [Ba].

Proof of Proposition 11.3.13. Combining (11.3.10) and (11.3.12) gives C2m(Z)= ~ tt,-- l~a+bcacat~ ! a b ~ m - a ~tz~t~ ] ~ m - b ~~z -- 1). a,b=O

Using Lemma 11.3.5, we can write a-b

C2m(Z)=

[Z]2m-a-b-$ (a--b--s)!s!2 ~

X Z (-1)~+b+'cacg O<_b<_a<_ms=O +

Y, O~_a
b-.-1 ( - 1)a+b+sCa-Ca(b-a-,v ~ s=O

1 +s)! [z]2,._~_b_ , (b_a_s_l)!s!2 9

Now set a + b + s = k, and collect terms in k, giving C2m(Z)=

2m Z (-- 1)k( Z

k=O

[a+bAk \a,b<~m

n A A) [ Z ] 2 m - k '

~abCa Cb

592

R. Howe and T. Umeda

where the t/~bare given by (13.3.14). But from (11.1.13) and the inversion formula for the calculus of finite differences, we know, in analogy with (11.3.10), that 2m

C2m(z)= E (--1)kCk[Z]2---k" Comparing this with the previous formula yields Proposition 11.3.13. (11.3.16) Corollary, We can write, for k <=m, _I CkA _---~Ck + Rk(Co, Ct ..... Ck- 1)

where R k is a polynomial with rational coefficients independent of m. Furthermore, C~ belongs to ~d(fll,) tk), i.e., the monomials l-] C~' occurring in R k saitsfy Y, ia i ~ k. i

i

Proof. From Proposition 11.3.13 and the fact that C~ = Co = 1, we see we can write C ~ = ~1 ( C k -

E

C.ACb~ --

~

~l.bC~ ~ ~ Cb ) .

a+b
a+b=k

a.b*O

Assuming the corollary is true for a, b < k, we see from this equation that it is true for k, so the conclusion follows by induction on k.

Remarks. a) It would be desirable to have more explicit formulas for the C~, analogous to Cape[li's formulas for the Ck's directly in terms of the Eij. b) The first few cA may be written as follows CoA= Co = 1 (11.3.17)

,t_1

C 1a = 2 -1C 1 1.

r~A_!r, "..-'2 - 2'-~2 _ •8 ".-~1 1

4

3/.-,2

4"-'1

1/.-

C3 - -2C3 - 8 C 2 C 1 "~-3-2C1 "~ C2 - - f f 6".-~1 - - 2 ~-~1

We may now use the formula of Corollary 11.3.16 to define elements CA of A~(flI,). Then clearly the formulas of Proposition 11.3.13 will be valid, so we can express the Ck in terms of the C~. Hence the C~ form a set of generators for ~r~(g1,). And since the Ck are stable, in the sense that their values on a given Offdepends only on D, not on n, the C~ will also be stable in this sense, since their expressions in terms of the Ck is independent of n. Therefore (11.3.6) describes Q,D(Ck ). Therefore we have the following characterization of the C~. (11.3.18) Proposition. a) The elements C~, for k<=n/2, are characterized up to multiples as elements of :L~q/(gl,) by the conditions i) C~ ~ ~ql(~l.) ~) ii) CA vanishes on Qo if D is a Young Diagram with columns of even length less than 2k. b) The elements CA, for k > hi2, vanish on Qff for all Young Diagrams with columns of even length at most n. Combining this with the characterization (11.3.5) of the FkA, we obtain (11.3.19) Corollary. (Capelli identities for skew-symmetric matrices). Under the action 2 of GL, on ~(A2(C")) we have 2(C~) = SkFkA 2(C~)=0

for appropriate non-zero scalars sk.

1 < k <-n/2 k>n/2

The Capr identity

593

Remark. If n = 2m is even, then det T can be non-zero. It is clearly an eigenfunction for the action of GLn. It is the square of the Pfaffian. The most naive version of the Capelli identity in this case would be a formula for the operator &a= (det T)(detd(T)) in terms of the 2(Eij)'s. By means of Corollary 11.3.18 this amounts to expressing Za in terms of the Fk'S. We observe that .~e can be given a characterization similar to that of the FkA. Precisely, A" is of degree 2m, and it annihilates all Q2 D- occurring in ~(A 2(C2'~)) such that b, < 1 (notation as in (11.3.6).. But is it easy to produce another such operator, namely 2(C~(0)C~(1)). That this has the characterizing property follows easily from (11.3.6). On the other hand, from (11.3.12) we see that 2(C~(0)C~(1))=;~(C2,(1)).Comparing top order terms then gives the formula (11.3.20)

(det T)(det ~(T)) = 2(Cn(1)) = det (2(Eij) + 6o~n- i - 1)).

This differs slightly from the parallel formulas for M n and $2(C~), which involve C,(0) rather than Cn(1). TurnbuU IT2] indicates another possible analog of Capelli's identity that holds for all n, in which the permanent replaces the determinant I-FZ]. Essentially identity (11.3.20) has recently also been found by Kostant and Sahi [KS]. 11.4. Case iv. On| GLI: This case is closely tied to the dual pair (On, SL2) [H1; H6]. Denote points of C n by

(11.4.1)

x=

2 .

LXnJ Realize On as the isometry group of the standard quadratic form 2 r 2 = r2(x) = ~. x~.

(11.4.2)

j=l

Then a basis for the Lie algebra on of O, is provided by operators

~ij = Xi~j-- Xjdi"

(11.4.3)

Here as in the previous cases we abbreviate ai = axi" Consider the Laplace operator (11.4.4)

A= ~ b2. .j=l

Clearly the operation by A on ~(C n) commutes with the action of O,, and so does the operation of multiplication. It is well known [H5] that if we set (11.4.5)

e+=ir22 e

iA2 h=[e+'e-]= J~=,xJ J + ~,

then these operators define a standard basis for a copy of the Lie algebra ~12; further the pair of Lie algebras form a reductive dual pair in the sense of [ H I ] (see Sect. 4a for this and the discussion below of spherical harmonies).

594

R H o w e a n d T. U m e d a

Denote by o*ff" the kernel of the mapping A :~m(C")-"~#m-2(C"). Let #(r 2) denote the algebra of polynomials in the variable r 2. The theory of spherical harmonics says that multiplication inside ~(C ~) induces an isomorphism of On-modules. (11.4.6)

Z ~'|

2)--- ~(C").

m=e

The ~ m are irreducible modules for On, with highest weight vectors z"; = (xl + ix2)".

The polynomial r 2 is invariant for On, but of course is an eigenfunction for the action of GL r Thus the two fundamental weights in this case are (11.4.7)

zl=xl +ix2,

r2 .

The invariant differential operator corresponding to z 1, which just generates C" as On-module, is of course the Euler operator gxjdj, which is just the infinitesimal generator of GLt. The invariant differential operator corresponding to r 2 is (11.4.8)

rZA = --4e+e - .

This can be expressed in terms of operators from ~q/(o, Oglt) as follows. The Casimir operator of o, is (11.4.9)

ego--

~,

E'~.

l
The Casimir operator of sI2 is (11.4.10)

r

+ 2(e+e - + e - e + ) = h 2 - 2 h + 4 e +

e- .

Because the algebras on and ~[2 form a dual pair there is a relation between the images of their Casimirs in ~ ( C " ) l-H6]. Denoting the homomorphisms from q/(o,) and ~(~12) to ~ ~(C n) by O, we have the relation tl 2

(11.4.11)

a('r

= o(~e,) + ~ - - n .

This may be verified by straightforward calculation. Combining (11.4.10) and (11.4.11) gives us the formula we want, since h - n is the infinitesimal generator of

fill. (11.4.12) r 2 A = h Z - 2 h - q ( c g o ) - ~ - + n =

h-

h+~

This formula is classical [KS]. 11.5. Case v. Sp2n| This case and the next 3 are related to the dual pairs ($p2,,Om) discussed in [H1]. This case is essentially trivial: the action of Sp2, on ~z(C2") is irreducible for all I. The only fundamental highest weight is xl, and the corresponding invariant operator is the Euler operator, the infinitesimal generator of gl,. 11.6. Case vi. Sp2n| This is an action on (C2")2. We take n > 2 ; otherwise we are in Case i with n = m = 2. We denote a point of this space by (01, v2), with v~e C "'.

The Capelli identity

595

Each v~ has coordinates

xtiX2i

(11.6.1)

vi= x:.i 9

Y:']. Yni [

The coordinates xj, yj are standard symplectic coordinates, and for Borel subgroups we use the usual upper triangular ones. The Lie algebra gl2 of GL 2 is spanned bythe elements (11.6.2)

E',,b = ~ (xj,~,,~b + y~,,Oy~b)

a, b = 1, 2.

3=1

If we add n6,~ to these operators, and supplement them with the operators (11.6.3)

e += ~ (xjlyj2-YilXj2)

e - = ~ (Oxj,Oyj2-OyjlO,,j2),

)=t

j=l

then we get a Lie algebra isomorphic to 04, the Lie algebra of an orthogonal group in 4 variable.s Let g denote the space of polynomials which are annihilated by e-. According to [H1], ~ is a direct sum of irreducible (Sp2, x GLz)-modules ~,1,,~, a t >~2>0, generated by highest weight vectors (11.6.4)

x ~ - " d e t ~ x ' ' xt2] "2. kX21 X22A

Furthermore, if ~(e +) denotes the algebra of polynomials in the function e +, we also know from [H1] and [-Sch2] that multiplication in ~((C2") 2) yields an isomorphism (11.6.5)

~ ( ( C 2n)2) =

E

~V'~"~|

e+ )

~1 > a 2 >:_0

of (Sp2n x GL2)-modules. From (11.6.5) we can see that the fundamental highest weights are (11.6.6)

x~,,

detF xal [ xz~

x~2], x22A

e+ 9

Let ~ , ~ + , and y,o denote the (Sp2, x GL2)-invariant differential operators I--

1

corresponding by means of Theorem 9.1 to xtl, d e t [ x1~ x12[, and e + respecLx21

X12._[

tively. Of course A~ is just the Euler operator, the infinitesimal generator of the scalar operators, and comes from the first Capelli element in ~q/(gI2). The subspace of ~2((C2")2) supporting the irreducible (GL2. xGLz)-module A2(C2")| breaks up under restriction to Sp2. x GL 2 into a direct sum of the r--

-'l

representation ~enerated by det ] x lx x l 2/ and the line through e+. Thus from ~ L X 2 t x22A C ase i we know that s + +.L#~o is the image of the second Capelli element C2 e.~q/(gl2).

596

R. Howe and T. Umeda

On the other hand, the Lie algebra 04 is isomorphic to a direct sum of two copies of ~I2, and, as described in (11.4.10), the operator .~0 =e+ e will contribute to the Casimir of one of the copies of 8I2. Since we can expect a relation between the Casimir operators of~p2 n and 0a, we should be able to express 2,0 as the image of something in ~'q/(~p2n~12). This is indeed the case. The precise formula is as follows. For ~P2~, the Lie algebra ofSp2 ., we take as basis the elements (described as operators on ~(C2~)) (11.6.7)

E/i = xid~, - yjar,

P/#+ --- x/~rj + x jar,

Pij = y/O~ + y j ~ .

From these, we form the Casimir operator (11.6.8)

c~,p= i.y=t ~ E/jEji+12(PqPo++P~Pi+)"

For 04 we take the basis described in (11.6.2) and (11.6.3). We set (11.6.9)

c~=(o.b~l(E'ab+n6,~b)(E'ba+nSab) ) --(e+e-+e-e+).

Then a straightforward calculation shows (11.6.10)

0(q~.) = 0(c~.p)+

2n(n-

1).

Here again we use 0 to indicate the appropriate homomorphisms from the enveloping algebras of ~P2, and of o4 to ~((C2n)2). Thus, if we set 2 (11.6.11) O(qfg[)= E (E',b+n6ob)(E'b,+n6ob), a,b= 1

which is the 0-image of an element, which we call the Casimir element of Lrq/(gl:), we can combine (11.6.9) and (11.6.10) to get (11.6.12)

e+e-=89

2(E'~ + E':2)+ 2n2).

Of course the Casimir element 0(Tgt) can be expressed in terms of the Capelli elements (cf. (11.1.8)). However, we will not bother to do this explicitly. 11.7. Case vii. Sp:,| a. This is analogous to Case vi, but more involved. When n > 3, the abstract Capdli property fails, making it, along with Cases xi and xiii one of three cases on the list (11.0.1) when the abstract Capelli property does not hold. We will assume n_->3 in this discussion, since n = 2 is covered by Case viii. We use notation analogous to Case vi. Our vector space is (C2n)3'~'~C2n~C 3. We 1.1se coordinates {x/,, Yo: 1 <__i~n, 1 =<j<3}. From [ H I I we know that the algebra @~(C2~| sp~" of differential operators commuting with Spz, is generated by a Lie algebra isomorphic to 06, which is spanned by operators (11.7.1a)

E'ab+nSab=~ (x~ax,b+ y/.dj,,b)+nSo~, l
(11.7.1b)

e ~ = ~ (XiaYib--XtbYia)

(11.7.1c)

e ~ = ~ (c3~,0y,,--a~,~y,o) 1=1

i=1

1 <=a
1 <__a
The Capelli identity

597

The operators E~b figuring in (11.7. la) are the standard basis of the Lie algebra gla of GL 3, the second factor in our group Sp2, x GL 3. The quadratic polynomials e~ generate the algebra of Sp2,-invariant polynomials on C2"| a. We will denote this algebra by J . Let g _ :(C2"| C 3) be the space of polynomials which are annihilated by the three operators e~ of (11.7. lc). From [H1] we can deduce that g is a direct sum (I 1.7.2)

.g =

y.

.,~',.",.",

where ~f,,,.,2,,3 is the irreducible (Sp2,,x GL3)-module generated by the joint highest weight vector ,,,,_,2det[xl 1 ,, -la2-a, [ x ' ' XI2 X13] a3

"'

kx2, x::J

det[::: x22 x23[ ' X32

It further follows from l-H1] that ~'(C2"| (11.7.3) ~(C2.| ~,

X33J

3) is a direct sum ,r .r

a~>a2>a~>0

where j = ~(C2"| sp:-. To further decompose ~(C2"| we need to know the structure of ~,,f~,,.~:,~3.g as a GL3-module. The polynomials e,b + span a GL3-module isomorphic to A 2(C3) ~ e~ ~,with D2 as in (11.1.2). For n = 3, the representation e~ ~is, up to twisting by determinant the dual of 0~'~ C 3, the standard module. That is (11.7.4)

0~3 ,~ =det3|

D~, ) 9

It follows that (11.7.5)

,,r

Z e~' m ~ Z det~'|176

n~O

m_>_O

*.

Using the (dual version of the) very simple special case of the LittlewoodRichardson rule [Mc] corresponding to tensoring with 0~'~ allows us to conclude

~'"'~"~| (11.7.6)

Z

E b2___~2-a3 Z e~-*''-~''-*''-~'''~*~'*~ bt_~r162

E

O~"'+b' *b''''+b' +~''''*b' *l'J"

ra>-_bl+b2

=

bt6al-a2

b2~12--~r3 bt~0

On the other hand, consider the polynomials

QI~---X11

Q2=detrX"X'2]

XI2

Q3 =det [ x 1 2 : ~X22 121

LX21 x22_] Xll tO' _ e 12 + Y-..2--

Q3= ~

(11.7.7)

Ix1, Q4= ~

IX21

,.,[:

l

LXal

X32

] x33J

XI2

X13]

Xil

gi2

Xi3 = x l l e 2 + 3 - - x 1 2 e t 3+ +x13e~2

Yil

Yt2

Yia.J

0

x12 x13]

0

X22

,

X231= rXl, X131 + __ FXll

y,,X",,2x"y,3 x'31j LX,, x,,Je'2 LX2I

X22.J

598

R. Howe and T. Umeda

It is easy to see these polynomials are all (Sp2 . x GL3)-highest weight vectors. Further, it is not difficult to check that by means of the polynomials Qj and Q) we can account for all the highest weights in the decomposition (11.7.6). Thus the list (11.7.7) consists precisely of the fundamental generators for ~(C2~| N.

Remarks. a) It follows from this analysis that the products in decomposition (11.7.3) are the full tensor products. b) If we recall the decomposition, analogous to (1.1.12) (see the start of the discussion of Case i) of # ( C 2 n | 3) under the action of GL2n "• GL3, we may derive from the description (11.7.6) of the GL3-module structure of the Sp2~ isotypic components of #(C2~@C3), the following branching rule (11.7.8) .,(P~,a:,#,)l -~ (7(~O~-b2-b3,~2-bl-b3,'#3-bl-b2) ~2n

ISpzn - -

t~2<-IJ1-#2

b3<#2--#3 b1+b2_~#3

where o ~ b'') denotes the representation of Sp2~ with highest weight (a, b, c, 0, 0 .... ). (It is isomorphic to any irreducible subspace of ~ . b , , . ) Let L#j, ] = 1, 2, 3 and L#j, j = 2, 3, 4 be the (Sp2n x GL3)-invariant differential operators corresponding to the Q~ and Q'~of list (11.7.9). Calculations like those of Case vi show that ~1, LP2 + L#~, and ~ 3 + L#6 are the images of the Capelli elements in L~q/(gl3). Also ~ can be expressed by means of the Casimir elements of q/(eP2n) and q/(gi3). However, there is no analogous way to find La~. We will show: (11.7.9) Lemma. The abstract Capelli property fails in this case: the natural homomorphism

is not surjective. Proof. To establish the lemma, we appeal to the Harish-Chandra Homomorphism IHu] which describes, for any reductive Lie algebra g, the action of.~q/(g) on finite dimensional modules. It implies that .~q/(gl3) acts on the representations Q~'" ~' ~ n+l by functions which are symmetric polynomials in c c ~ + - - ~ - i. Also, ~eq/(~pg~3 acts on the representations a~2#,''#:'a~) by functions which are symmetric in (Pz + n + 1 - 0 2. As we have seen (11.7.6), the decomposition of ~(C2"| 3) under Sp2, x G L 3 consists of the representations

(11.7.10)

~'2n'V('81'#2' J/3)r',0' k'3 +b2+b3"#2+ba+b3.P3+b~, + b2)

Thus, if we act by ,~fq/(~p2,~gl3) on these representations, we get functions which are symmetric in (//~+ n + 1 - 0 2, and symmetric in/~1 + b2 + b3 + d - 1,//2 + b i + b3 n+l + d - 2, and/~3 .+ b~ + b2 + d - 3 where d = --~---. Set//'~ = 1~+ n + 1 - i. Then we get symmetric functions in the//~2, and in//'~ + b2 + b3 - d,//~ + b t + b3 - d, and//~ + bt + bz - d. We may translate these last variables by d to obtain functions symmetric in the//,2 and in fl~ + bz + b3,//~ + b~ + b~, and/?~ + b~ + b2. Thus we get a graded algebra, graded by total degree in the variables fl'~ and b~, and this algebra has generators of degree 1, 2, 3 and 2, 4, 6.

The Capelli identity

599

On the other hand, the algebra ~ ( C 2 ~ | C3)sp~"• OL~= ~r is naturally f'tltered by the usual notion of degree of differential operator, and it is clear that the action ofL,e e ~1 on the representations (11.7.10) is via a polynomial in the fl~and b~whose total degree is at most the degree of f f as differential operator. Comparing these two results, and looking at degree 3, we find that &r~'(ep2,~gla) can account for at most a 8-dimensional space of polynomials of degree less than or equal to 3 in the fl~and b~, but ~r provides at least a 9-dimensional space of polynomials of degree at most 3. Therefore ~(qg(ep2,OgI3) cannot map onto ~r and the lemma is proved. 11.8. Case viii. SP4@GLm: Here the space supporting the action is C4| ~. We use coordinates {x~j,y~j; i = 1, 2,1 < j < m}. It is convenient to analyze this action in terms of the branching rule from GL4 to Sp4. This can be found in [KT]. However, it may also be deduced fairly easily from the calculations of Case vii. If we take m = 2 rather than m > 3 in Case vii, then the only modification we need is to omit the polynomial Q3 from the list of generators. This means the harmonics are the span of the ~ f ~ " ~ with ~ >~2 > 0. This results in the branching rule

Q(flz,fl2,

f13, O} l ,-~ ISP4 =

4

O.~fll-b2-b3,#2-f13+b2-b3)

v/ . b2_~ min(~ 1 - ~ 2 , P3 ) b3 <=p 2 - p 3

Notation is as in (I 1.7.8). Since we can obtain the general irreducible representation of GL4 by twisting Qt4Pl,#2'a~'~ with a power of the determinant, which is trivial on Sp4, we can write the general branching rule from GL4 to Sp4 as follows. (11.8.1)

•4( f#2,f l#3,fl4) , [Sp4~-'= b2 ~ min(fll ~,,- f12, .83 - ~4) 0"{4ffl--b2--b3 --f14" ~ 2 - f l 3 q ' b 2 - b 3 ) b3 _-
=

2

O.(4at,,ce2),

t/l -<~el +~2_-< ~2 ~ O ~ a l -~2=
where

't2 =P, + P 2 - P 3 - P ,

'tl =P,-P~ +/33-~,

,to- 1(/3, - f12)- (f13 - f14)].

We know from Case i that the decomposition o f : ( C 4 | GL4 x GLm is

m)under the action of

where D = ( a l , a 2 . . . . ) is a Young Diagram of depth at most min(4, m). Combining this with the branching rule (11.8.1) gives us the decomposition ~(C4

(11.8.2)

O.(4az-b2-b3-a4,a2-a3+b2-b3) | (al,a2,a3,a4)

{ ~ C m) :~" b2_~ rain(at - a2, a3 - a4) b3 ~ a 2 - a3

~.(at t, ~2)/f~ ~(ct 1 + b 2 +b3 + a 4 , ~ 2 + b l + b 3 + a4,bl + b 2 + a 4 , a4) u4 ~ ~m ~1 >_-~2_>--0 b2 ~ { 2

t

600

R. Howe and T. Umeda

We must take a~ = 0 if m = 3, and a# = b ~= b~ = 0 if m = 2. It is routine to check that this decomposition can be produced by the simultaneous highest weight vectors Q2 = det [ xl 1 x121 LX21 X:Z2I

Q1---~Cll

Q~=detlX~21

LY:zl Y22

Q =det[ x11

{11.8.3)

Ix1,

LY11 Yl2J x12 xls l

0

Q4= ~det[X~l

x22Xt2 x2~/xl31 Y23J

LY2t Y221 [ x l l x12 x13 x~r

0 x22 x23[ Qr

xz2 x23 x2r

x,31 lYe1 Y22 Y23 Y24/ Y~I Y~2 Y~3J Ly~ y12 y13 Y~4J Thus these polynomials must be the fundamental B-eigenvectors. Let ~1, ~z, ~3, ~ , La~ and ~4 be the (Sp4 x GL,,)-invariant differential operators associated to the various Q's. From Case i we can see that if Ci are the CapeUi elements for GLm, and # denotes the action of GLm on ~(Cr174 then

,=1

[00

xil

xiz

Further Ze~ can be expressed in terms of Casimirs, as illustrated with Case vi. Thus the interesting operator here is ~ . It can be determined as a polynomial in the Ci, 1 _~i__<4, and the operators from .~ql(~p4), by means of the characterization (11.0.3). However, the result does not appear to be especially enlightening, so we will not give it. Case ix. Spin7| Recall [H1; H5] that the Clifford algebra C2. of a symmetric bilinear form (over C), can be represented on A(C") in such a way that ~o2n, the Lie algebra of the Spin2. group, consists of the operators

89

(11.9.1)

^i)

hi ^j

JtJj

l<=i,j<=n, 1 <_i<j
Here A ~indicates wedge product with the i-th standard basis vector of C ~, and J~ is the corresponding "differentiation" inside the exterior algebra A(C". See [H1; H5]. The even degree exterior forms ~ A21(C") = A~""(C"), and the odd-degree l>__o

forms ~ A21+I(C~)=A~

are invariant under the operators (11.9.1), and

1~o

constitute the two spin modules for ~02.. Take n=4, and consider the subalgebra of ~os which stabilizes 1 +el A e2 A es ^ e4 ~A""(C4) 9This consists of the operators

^~Jj

(11.9.2)

(1~i~j~4), hu= A(Ji - All j (1 =
Of course, only three of the hu can be linearly independent. One can check that this algebra is isomorphic to ~aT. (The ^ d j and h o span a copy of s14, which is

The Capelli identity

601

isomorphic to so 6, and the ^i Aj+JkJl span a copy of the standard 6-dimensional module for s%.) The module A~ 4) is a copy of the spin module for the copy of ~07 described in (11.9.2). Consider in this algebra the subalgebra which stabilizes e4 + el ^ e2 ^ ca. It is the 14 dimensional algebra spanned by the operators h12=elJl -e2J2 (11.9.3) AiA4-t-J/Jk+ ^ J 4 ^j ^kd-]i]4 - ^4J~

hza=e2J2-eaJa,

eiJ J (1 <=i=~j<3), (i,j,k a cyclic permutation of 1,2,3), (i, j, k a cyclic permutation of 1, 2, 3).

This is easily checked to be a copy of the exceptional Lie algebra G 2 [ H a ] . The elements h12 and he^ span the Cartan subalgebra. If we further ask for the stabilizer in the G 2 of (11.9.3) of the vector e 4 - e ~ ^ e 2 ^ e a (which is the same as asking for the stabilizer in ~o 7 of both e 4 and e~ A e 2 ^ ca) we find it consists of the span of the 8 operators

(11.9.4)

hi2 = e l i 1 - e e J 2 hza = e 2 J 2 - e^J^ eiJ j (1 <--_i4:j<3)

from the first two lines of (1.1.9.3). This is a copy of ~la, which is acting on A 1(C4) = C 4 ~ C a ~)C by its standard representation plus a trivial representation, and on A3(C 4) by the contragredient action. These observations support the following conclusions (which are known [I; Hal). (11.9.5) i) Spin7 x GL1 acts irreducibly and prehomogeneously on A~ and the stabilizer of a generic point is G2. ii) G 2 • GL 1 acts irreducibly and prehomogeneously on the 7-dimensional orthogonal complement (with respect to the ~o8-invariant inner product on A~ of its fixed vector, and the stabilizer of a generic point is SL3. iii) The 6-dimensional representation of SL a on the orthogonal complement of its two fixed vectors in A~ 4) is the direct sum of C a and (C3) *, and SL3 x GL1 acts prehomogeneously on it. We may use these facts in conjunction with part vii) of Theorem 9.1 to describe ~(A~ (We will use them later in Case xii also.) According to Theorem 9.1.vii and Corollary 4.5, there are isomorphisms

~,~(Aod~(C'))sP"7-~~(A~

~ ~(C)|

G~.

Since G 2 x GL 1 acts prehomogeneously on C 7, we see that #(C7) G2 must be generated by the restriction to C 7 of the ~os-invariant inner product on A~ Thus #~(A~ sp~n7is generated by two elements, one of degree 1 and one of degree 2. The degree 1 element is of course the Euler operator, and the degree two generator comes from the invariant inner product, and must be constructed from this and the Casimir for ~07, as in Case iv. To compute the relation precisely, rather than compute the relevant operators explicitly in some coordinate system, we will compare them via their eigenvalues. The standard formula [Hu, p. 134, Ex. 4] for the action of the Casimir operator specialize to show that on the mth Cartan power (cf. [Hel, p. 5451) of the spin representation the Casimir ~r has eigenvalue 3re(m+3). The Euler degree

602

R. Howe a n d T. Umeda

operator of course will have eigenvalue m on the realization of this representation inside ~(A~ Thus if r oT)denotes the image of ~#~o7in 3~176 we see that the combination of E, the degree operator, and o(c~,~ which annihilates the powers of the degree I fundamental highest weight is 3E(E + 3 ) - cr(C~,o7).This has eigenvalue 3 . 2 . 5 = 30 on the invariant quadratic polynomial. On the other hand, if we set up coordinates as in Sect. 11.4 for n = 8, the operator r2A will have eigenvalue 16 on r 2. Hence we have (11.9.6)

r2A =~E(E + 6 ) - ~-~s(c~,oj

if rZA is the operator on the left hand side of formula (11.4.12) for n= 8. 11.10. Case x. Spinlo| GLI: This action can be realized on AeVen(CS),as described in Sect. 11.9. This case comes from the Hermitian symmetric space Er/Spinlo, and is treated in [J]. There are just two fundamental highest weights; the one of degree 1, which of course generates the spin representation; and one of degree 2, which generates the standard 10-dimensional representation of SOlo. (We remark that the numbers 146 and 136 on p. 79 of [J] should be 136 and 126 respectively.) Again we can predict that the commuting algebra of Spin~o in 3 ~ ( A .... (C5)) will be generated by theEuler degree operator and the Casimir of ~ol o- The order 1 generator is as always the degree operator. Let us call the order 2 generator ~ . To get a formula for ~ , invoke remark (11.0.3) and argue as follows. There are four ~a~o modules in 3~J(ACV~ j < 2. Also we have four Spin~o-invariant operators of degree at most 2, namely 1, the identity operator; E, the Euler degree operator; E2; and 0(C~,o), the image of the Casimir operator of ~O~o. The operator L~ is characterized by properties (11.0.3), and we use the characterization to compute it. Let Qt, Q2 be the two fundamental highest weight vectors. Then obviously. (11.10.1)

E(QT'Q"; 2)= (m 1+ 2mz)QT'Q'~ 2.

On the other hand, from the formula [Hu] for the action of the Casimir operator, and our knowledge of the highest weights corresponding to Q1 and Q2, we find (11.10.2)

O(,o) (Q1) =zml(ml + 8)Q'~~ 0(~,o) (Q~'~)= m2(m2 + 8)QT~9

Manipulating formulas (11.10.1) and (11.10.2) shows us that, up to multiples (11.10.3)

Ae = S E(E + 8 ) - O(c#,,).

11.11. Case xi. Spins| Igusa [I] has shown that the stabilizer of a generic point in the spin module for Spin9 is Spirt 7. The restriction to Spin7 of the spin module for Spin 9 has three components: the spin representation for Spin 7, the standard 7 dimensional module, and the trivial representation. We know the spin representation and C 7 both admit invariants of degree 2. In particular, they are self-dual. Following our discussion of Spiny in Sect. 11.9, and of the standard representation in Sect. 11.4, we know that the polynomials on the spin representation on the one hand, and the polynomials on the standard representation, have no non-trivial ~o7-modules in common. Hence we can apply Corollary 4.4 to conclude that .~ + (Sping) has rank 3, and that ~(Spin9) N has one generator of degree 1 and two generators of degree 2. The fundamental generator of degree 1 of course is the highest weight vector for the spin module of Spin 9. The fundamental generators of degree 2 come from restricting the standard (10-dimensional) module

The Capelli identity

603

for ~01o to 309. The result is of course the standard module for s%, plus a trivial representation. That is, Spin 9 admits an invariant inner product. On the other hand, the algebra ~o//(~09Ggl 0 has only one generator of degree 2, the Casimir operator for ~o9. Thus unless the map (10.3) specialized to this case involves some subtle cancellations, the abstract Capelli problem must have a negative answer in this case. We show this is true. (11.11.1). Lemma. When ~:509{~)~l 1 and V is the spin module for so 9, the map (10.3) is not surjective. Proof Let wl be the highest weight of the standard module for ~09, and let w4 be the

highest weight of the spin module. The theory of the Harish-Chandra homomorphism says that the eigenvalues of elements of ~q/(~Og) on the representation with highest weight lwl + m w 4 is a symmetric polynomial in the variables

m+3 Make the substitutions/2 = ~ and 2 = l+ 89 + 7). Then we find are dealing with symmetric functions in the variables 22

(p+l)Z

pz

(#_l)Z.

Let us compute the elementary symmetric functions at in these variables. They are o1=22+3#2+2 a2=22(3#2+2)+ 3#4+ 1 cr3 = 22(3/24 + 1) +/?(#2 _ 1)2 a4 = (22#2(# 2 - 1)z. We observe that al - 2 is homogeneous quadratic in 2 and #. The content of the lemma is that any polynomial in the 0-~which is homogeneous quadratic in 2 and # must be a multiple of0-1 - 2. To see this we will use a geometric argument, for which we prepare by manipulating the at in order to obtain a generating set as simple as possible. First, replace the ai by the slightly simpler set zl =0-1--2=22

+3/22

z2=

0-2 - 13 -- 2zl =/22(22 + #2 _ 2)

"ca = 0"3 - - 'C1 - - 'C2 = / 2 2 ( 3 # 222 -{- # 4 _ 22 _ 3 # 2 )

"c4 = O"4 = 22/22(# 2 - - 1)2.

Second, change variables: put 2 2 +2# 2= a and #2= b. Then 9cl = a

"c2= b ( a - 2 b - 2)

Now we check that

'ca = b ( 3 a b - 8b 2 - a)

4"C4='Ct'C3_3"C~+.C1"C2_6.C3

"c4= ( a - 3 b ) b ( b - 1)2. "

Hence it suffices to show that we cannot find a polynomial in 'cl, 'c2 and 'c3 which equals b. This is clearly equivalent to showing that the pullback map 'c' : ~(C3)--,~'(C 2) defined by the mapping 9c:(a, b)-~(zl(a, b), "c2(a,b), 'ca(a, b))

is not surjective. We show this by using the following

604

R. Howe and T. Umeda

Criterion. If the pullback map ~' : ~ ( C 3 ) ~ ( C 2) is defined by a mapping r : C 2--}Ca, -c(a, b)=(zl(a,b), z2(a, b), xs(a,b)) is surjective, then the tangent vectors

D=~= L Oa

Oa

OaJ

D~ = k ab

ab

ab J

must be independent for all (a, b)~ C 2. This follows by differentiating a putative set of equations

a=P(~,%~3)

b=Q(~l,%%)

with respect to a and b, by using the Chain Rule. For our given T, we compute D~t=[1

b

3b2-b]

D~t=[0

a-4b--2

6ab-24b2-a].

We see that Dbz(3, 1/4)=[[0 0 0], contradicting the Criterion. This proves Lemma 11.11.1. 11.12. Case xii. G2| The action of Gz on C 7 has been studied by several authors, especially Haris [Ha] and Schwarz. From these works, or from Sect. 11.9, we know that the stabilizer in G 2 of a generic point in C 7 is isomorphic to SL 3, which acts on C 7 by a direct sum of its standard 3-dimensional representation, the dual of this, and a trivial representation. We see that this SLa action has 2 fundamental invariants: the fixed vector and the pairing between C a and (Ca) *. Thus ~ +(C7) will be of rank 2, and the fundamental highest weights are of degree 1 and 2. The degree 1 highest weight is of course just the highest weight of C 7 itself, and the degree 2 highest weight is just the G2-invariant inner product. The fundamental generators of the invariant differential operators are similar to those in the case of SpinT: the first order generator is] as always, the Euler degree operator, and the second order generator is some expression in the degree operator and the Casimir (of G2). The expression is determined by the criteria (11.0.3). Again using the standard formula for the action of the Casimir [Hu] we can compute the second fundamental operator to be

( 11.12.1 )

2E(E + 1) - e(~o:).

The parallel with formulas (11.4.12), (11.9.6) and (11.10.3) is clear. We use the same convention as in those formulas, denoting by Q(cgo~)the image of the Casimir operator of G2 in ~ ( C 7 ) . 11,13. Case xiii. E6| This is an interesting action, but we have relatively little to say about it. According to [J], there are 3 fundamental highest weights, of degrees 1, 2 and 3. The degree operator and the Casimir ofE 6 may be used to express the fundamental commuting differential operators of degrees 1 and 2. However, the operator of degree 3, which is the most interesting because it corresponds to a cubic invariant for E6, is not in the image of ~~ One would expect this because the degrees of the fundamental generators of :~q/(E6) are 2, 5, 6, 8, 9, 12 and, so construction of an operator of order 3 would have to involve some tricky cancellations. That these do not occur has been checked by

[Heal.

The negative answer to the abstract Capelli problem in this case means that the strategy to compute b-functions via a Capelli identity must fail in this case. For

The Capelli identity

605

more details on computing the b-functions for this action see I-RS]. In [KS] a somewhat looser sort of identity analogous to what we have been here calling a Capelli identity is established by the reverse procedure: by computing the b-function for the operator, then comparing with elements of Y'q/(E6~I~). Of course here the element which maps to the fundamental commuting operator of degree 3 must lie in some ring of fractions of .~q/(E6~flll).

Orbits and G-invariant ideals 12. Let G be as in Sect. 2 and I be a G-invariant ideal in ~(V). The theory of highest weight tells us, as sometimes quoted above, that I is recovered from I s = I c~(V)N: I is generated by I N as a G-module. Considering that the highest weight vectors are determined by the corresponding characters, I is determined by (12.1)

"i+(I) = {V'; Q,~~ : } 9

As we saw in Sect. 2 (see also Theorem 9.1), .~+(V) is a free abelian semigroup generated by P_~+(V), so that we can define a natural order < on it: Ipl <~P2 if ,~i- 1~2 is expressible as a product of primitive highest weights. Since I is an ideal, +(I) has the following upward hereditary property with respect to this order: ~x e,4+(I) and ,pl
Proof Obvious from the definition of primeness. In fact, if a minimal element in .~ + (1) decomposes non-trivially, then the corresponding A-eigenvector is accordingly factorized. One of those factors, which is also an A-eigenvector, must belong to L This contradicts the minimality. (12.3.) Corollary. The number of G-invariant prime ideals is finite. Besides the order < , let us introduce two new (pre-)orders or "closure relations", in ~ +(V) by using the sets of the form .~ +(I). For lp 1, ~P2~ ~ +(V), denote by lpl ~ ~P2 (resp. ip 1 ~<~P2)if ~01e ~ +(I) implies ~P2e ~ § (I) for any G-invariant ideal / (resp. for any G-invariant prime ideal 1). Define ~(v2) (resp. ~ ( v 2 ) ) as the ideal generated by Y~ (resp. the radical of 30P)), which is seen to be the smallest G-invariant (resp. G-invariant semi-prime) ideal of ~(V) containing Y~. Then we have the equivalent definitions: (12.4a)

~Pl%~P2 if and only if ~(~pl)~(~p2),

(12.4b)

~p~~I]) 2 if and only if ~ ( l p l ) 2 ~(~P2).

The part a) is obvious. For b), take the associated prime ideals Pl ..... Pr of ~ ( ~ p ) (see e.g. [ZS, Chap. IV, Sect. 5]). Since p i c~... rip, = ~ ( ~ ) , we have only to show that each Pi is G-invariant. From the uniqueness of the associated primes, we see that the action of G causes a permutation on the set {p~ ..... p,}. Any G-orbit in it must be a one point set, because, by assumption, G is connected. Clearly ~ < ~P2 =:" ~Pl~ v22 =~ ~Pl ~<~P2- Proposition 12.2 tells us that the order ~< is determined by the restriction to the subset P.~+(V). We will determine the ordered structure (P_~+(V), <~) in relation to the G-orbit structure of V (see Proposition 13.11 below). The order ~ is sometimes easier to calculate directly than ~<. We observe

606

R. Howe and T. Umeda

(12.5) Proposition. If V1%~2, then degtp 1_-<det~p2. Moreover if deg~pl =deg~2 , then Vt % ~2 can occur only when ~ol = tP2. Therefore the pre-order % is an actual order. Here degw means the homogeneous degree of the representation Y~ in ~(V).

Proof. The degree of any irreducible component of the G-invariant ideal 3(ip) is greater than or equal to deg~p, because every irreducible component of ~(V) is homogeneous. The second assertion is obvious because multiplication by a nonconstant dement in ~(V) raises the degrees strictly. About, <, we see (12.6} Lemma. Let Qv, be an irreducible relative G-invariant. If tp < tp' and ~p~ tp', then deg~ < deg~p'.

Proof. In this case, the G-invariant ideal ~(v:) is the pt'incipal ideal generated by Q~. Since Q~ is irreducible, ~(~) is already prime. Then the assertion is clear. We give here some examples about the order 4 , especially on PA+(V).

GLn|

(12.7)

S2GL~,

As we saw in Sects. 1 and 11.1, for GLn|

A2GLn

the primitive highest weight vectors

i,~,., tl 2 ... tl.i1

are

(_ tii ...........

tii _]

We show v?i%~P~+~(1 _<_i
1,

SpinT@GL1,

Gz|

Spinlo|

SpEn@GL1

These cases are dear, because at most two primitive highest weights are here. And ~a~~tp z is immediate from the definition, where ~ (i = 1, 2) is the primitive element of degree i. (12.9) Sping| There are three elements in Pft§ whose degrees are 1, 2, 2. Denote those elements by ~pt, v:2, u By definition and Proposition 12.5, we see Wx~lp 2, v: ~~ tp~ but neither W:,< ,,, ~p:, nor u ,.< ~ ~P2can hold. Thus (P~ +(V), ~ ) is not totally-ordered. We observe, however, that (P,~+(V), <) is totally-ordered from the orbit structure (see Proposition 13.11 and (13.3)). Let us explain the situation briefly. We can speeitiy the elements v)2, ~p~as follows. The symmetric square S2(V*) decomposes as Yd ~ Y~0) Y~,~,where dim Y~ = 126, dim Y~ = 9, and dim Yr = 1. Actually Y~ ~s the vector representation ofSO9t~GL ~. And Y~ restricted to Spin 9 is just the trivial representation. We will see later (13.13) that ~(~P2) - ~$(~P~), so that W2< ~P~"In this case, the radical ~ = @~0P2) of the ideal ~ 2 ) is the ideal generated by Y~,~and Yr which is prime. 9

(12.10)

Sp2,|

Sp2~|

$p4|

(m>4)

The Capelli identity

607

In these cases there appear duplications in the degrees of the primitive highest weights (see Sects. 11.5-11.8). Therefore (PTI+(V),~) cannot be totally ordered. The case Sp2,| gives us even a non totally-ordered example of (P.~+(V), 5) (see (13.14) below). The other interesting case is Sp4| where ~< on PJ+(V) is actually not an order. This means that ~p~<W' and ~p'~<~ do not necessarily imply = ~'. We can see this by comparing the cardinalities of the set of G-orbits and of P.~§ This kind of degeneration also occurs in the case of Sp2,| (see (13.14), (13.15) for more details).

E6|

(12.11)

1

There are three primitive highest weights ~Pl, WE,I/)3with deglp i = i (see Sect. 11.13). According to [-J], as E6-modules we see Yw2is dual to Y~, and Y~o~is trivial. In other words, "

Therefore Q~0~appears in the ideal ~(~2), so that ~P2%~v3- This proves that on P~+(V), both ~ and < are totally-ordered. From Lemma 12.6 we see that tp2 ~ ( ~ 3 ) , so that < is an actual order on P.~+. 13. For multiplicity-free actions, the primitive highest weight vectors are geometrically described. They are the irreducible polynomials that vanish on some codimension 1 component of the complement of the open B-orbit. Here B is the Borel subgroup of G acting on V prehomogeneously. This is just a fundamental fact in the theory of prehomogeneous vector spaces, because the primitive highest weight vector is nothing but the primitive relative B-invariant. (See [S, Theorem 1; SK, Sect. 4, Correlation 6, also Sect. 3].) In this section, we will take a closer look how the G-orbits reflect the G-module structure of ~(V). The set of G-orbits in V, which we denote by s has an ordered structure naturally induced from the closure relation: for @~, (_92e ~(V), we define (0x ~<@2 if (9a ~ ~2 = Zariski closure of (92. The following list is the explicit closure relations for our irreducible multiplicity-free actions. Out of 13 series of those actions, only one case (Spz,,| turns out to be non totally-ordered. (13.1)

GL,,|

S2GL,,

AZGL,

These cases are similar, because the orbits are parameterized by the rank of matrices. GLn| GLm: (90~ (91<.'.. ~ (groin(re,n),

S2GL,: (9o<(9~ < ... ~(~,, A2GL,,: (90 < 02 < . . . ~<(92t,/21Here (9~ is the orbit whose matrix rank is i. (13.2)

O,,|

,

Spinv|

,

G2|

In these cases, the orbits are (0}, (q = 0}\{0}, (q ~ 0}, where q is a quadratic relative G-invariant. This is clear for O,,| See [O, Proposition 4] for Spiny, and [Ha] for G2. (13.3)

Spin9|

The orbits are {0}, ~0~,(gz, @3. Here (9~ is the open orbit defined by a quadratic relative G-invariant q, and {q=0}\{0} breaks into two orbits 0~, d~2 of

608

R. Howe and T. Umeda

codimensions

5

and

1 respectively.

(See

Spinlo|

(13.4)

II, Proposition5].)

Clearly

I

The orbits are {0}, (91, r Here (92 is the open orbit and 61 is of codimension 5 (see II, Proposition 2]). So dearly {0} < r ~ r (13.5)

Sp2n@GL1,

Spzn@GL2 ,

Sp2n@GL3,

Sp4|

m

In general, identifying the representation space as the space of 2n • m matrices, we can determine the orbits for Spzn| GLm by two data: (a) rank and (b) isometry type (Witt theorem, cf. [Ja, Chap. V]). Those data are described as (a) r = rank X and (b) s = rank XtJX, respectively. Here X is a 2n • m matrix, and J is the alternating matrix defining the symplectic group Sp2n. We have obvious limitations like r < min(2n, m); s =<m; s < r. Also the integer s must be even, and s > 2 r - 2n. Let us simply denote by (r, s) the orbit with these two data. Then the closure relation is given by (r, s) ~<(r', s') if both r <-r', s <=s' hold. Returning to our cases, we see the following

Sp2,| Sp2,|

(0, 0)~<(1, 0) (0, 0) ~<(1, 0) ~<(2, 0) ~<(2, 2)

Sp2,|

(0, 0) ~ (1, 0),,.<(2, 0),,~<(3, 0) ~A ~A

(2, 2) ~<(3, 2) SPa|

(0, O)~<(1, O) ~<(2, O) ~<(2, 2) ~<(3, 2) ~<(4, 4)

In the diagram for SP2,| when n = 2. (13.6)

we should drop (3,0), (3, 2) when n = 1, and (3,0) E6(~GL 1

According to Proposition 13.11 below and the structure of the primitive highest weights P.~ +(V) (see Sect. 11.13 and (12.11)), there must be four orbits, which are totally ordered. In fact, the observation in (12.11) assures that no degeneration for < occurs in P~+(V). We shall now examine the relation between G-orbits and G-invariant ideals of ~(V). By definition, to give a closed G-invariant set in V is equivalent to give a downward hereditary subset in s Here we call a subset or downward hereditary (parallel to upward hereditary in Sect. 12) if ~ ~ or and (9' ~<(9 implies dT'e or. Since G is connected, so is every G-orbit. An irreducible closed G-invariant set is of the form ~, the doseure of a single G-orbit, which corresponds to a downward hereditary set generated by a single set {(9}. Let us introduce a kind of "pairing" between the two sets ~+(V) and (9(1/) as follows: for ~ ~ ~ + (11) and d~~ ~3(V). 0,

(13.7)

Qp, O>=

1,

if Q~ vanishes on (9, otherwise.

With respect to this "pairing", consider the annihilators: (13.8a)

~p• = {(9 e D(V); = 0},

(13.8b)

r = {~, ~ ,t+(v); <~,. ~> =o}.

It is clear that ip• is downward hereditary in s and that d~~ is upward hereditary in/] +(V) in any sense of three (pre-)orders introduced in Sect. 12. For Y_~~(V) and

The Capelli identity

609

S ~ V, define the common zeros of Y and the annihilator ideal of S: (13.9a)

C ( Y ) = {re l~ y(v)=0 for any y~ Y},

(13.9b)

J(S)= (P~(V);

P vanishes on S}.

If Y is a G-module, then ~e-(y) is G-invariant. Also if S is G-invariant, then Jr(S) is G-invariant. (13.10) Proposition. We have the following

(1)

~:(Y~)= U o,

(2)

jW)=

(3)

y. y~,

jr((9) N= jr((~)c~:(V) N= ~,~,~CQ~o.

Proof The assertions (1) and (3) are immediate from the definitions. And (2) follows from (3), because J((9) is generated by jr(O) N as a G-module. Since tP is connected, J(tY)is a prime ideal. From the definitions of .~+(I) and 0 -t', we see

~i +(jr(e))= ~ . As we saw in Proposition 12.2, .~ +(J(r is the upward hereditary set generated by P~] +(jr((9))=.,] +(J((9))nP.~+(V). On the other hand, the Hilbert Nullstellensatz says ~ ( J ( S ) ) = g and I=jr(r for any subset S of V and semi-prime ideal I. Restricting ourselves to the case where S and I are G-invariant, we conclude (13.11) Proposition. The annihilator operation gives us two reciprocal orderreversing maps between the set of downward hereditary subsets of (9(V), ~<) and the set of upward hereditary subsets of (P_~+(V), ~<). (13.12) Corollary. The number of G-orbits is at most the number of the upward hereditary subsets of the finite set P.~+(V), so that it is finite.

Remark. The finiteness of the nubmer of G-orbits for multiplicity-free actions was already proved by [Se] and also [K]. + This correspondence gives us the structures of P• (V) and 9(V) more clearly. First of all in case s is totally ordered, h9 ( V ) - 1 __<~P.4 +. (The annihilator of the open G-orbit is the empty set.) And if the closure relation ~< does not degenerate on P~+(V), the cardinality of singular G-orbits and that of the primitive highest weights are the same. This is true for almost all cases. In the rest of this section, as a supplement of the above discussion, we consider some interesting examples. (13.13)

Sping|

,

Spinlo|

We observe the relation between those two cases. To avoid confusion, we will put the superscript tloj or t9), if necessary, referring to the case Spinlo or Spin 9 respectively. For Spin 1o, the symmetric square Sz(V*) breaks into two irreducibles as-f~'~176~-~2 . .l-l~r~,~.~,.=, .... 1,2~is w thede~ree e ,~i ._• iDrimitivehighest g h t ,. . In general, the orbit {0} is corresponding to the irreducible representatmn V* = ~'(V*)= Y~. Considering this, we see that y~0) "IP2 generates the prime ideal corresponding to the

610

R. Howe and T. Umeda

orbit (9~1o) ofcodimension 5 (see (13.4)). Actually y$~o)is the 10 dimensional vector representation for SO 1o| GL1. And the restriction of this representation to Spin 9 x GLt breaks further into two: y~O,= y~91~ y$~). Here Y~) is trivial o n Spill 9 and y~91is the vector representation for S09@GL 1 (see (12.9)). Now consider the zeros of two ideals ~ ( y ~ o))~ ~(y~91). Then we see '-'t/~lo)c-~,~1/~(9).Since the closure of the (Spin~o x GL1)-orbit 0~1~ is of course (Spin 9 x GL0-invariant, which is of the same codimension 5 as ~91, it must coincide with ~91. From this, and the
Spz.|

(13.141

3

There are six primitive highest weights, and five singular orbits here. Let us give the "multiplication table" in the sense of (13.7) explicitly. [-See (11.7.7) and (13.5) for notation.] P~\orbits

(0,0)

(1,0)

(2,0)

(3,0)

(2,2)

(3,2)

Qt Q2

0 0

1 0

1 1

I 1

1 1

1 1

Q3 Q~ Q~

0 o o

0 o o

0 o o

1 o o

0 t o

1 t 1

Q~.

0

0

0

0

0

1

This shows that degeneration occurs in this case (for Q~ and Q~).

Sp4|

(13.151

There are six primitive highest weights, and five singular orbits. As above, we give the explicit table here. [See (11.8.3) and (13.51 for notation.] e/t +\orbits

(0, O)

QI Q2

Qa Q4 ~2~ (2:,

(1, O)

(2, O)

(2, 2)

(3, 2)

(4, 4)

0

1

1

0 0 0 o o

0 0 o o o

1 0 0 o o

1

1

1

1 0 0 1 o

1 1 0 1 t

1 t 1 1 1

14. Among the various ingredients in Theorem 9.1, what we have so far not discussed in relation to G-orbits is the H-invariants. In Theorem 9.1, H* was the generic stabilizer, i.e., the stabilizer at a point in the open G-orbit. In this section we discuss in general H*-invariants, where H* is the stabilizer at a point in a G-orbit (9. (14.1) Lemma. For any G-orbut (9, H* is spherical, i.e., there exists a Borel subgroup B such that BH* is open dense in G.

Proof. Consider the attine variety ~ = Zariski closure of (9. Then the action of G on is multiplicity-free, because the ring of regular functions C[-~] on ~ i s a quotient of ~(V). Then the criterion in [VKI, Theorem 2] or [K, Lemma 13.8] gives us the assertion.

The CapeUi identity

611

(14.2) Proposition. Let r* : # ( V O V*)o ~ ( V * ) n ~ be the homomorphism defined as

(r'P) (v*) = P(v~, v*)

(P ~ ~ ( V ~ V*)~).

Here v~(9 and H* is the stabilizer of v~. Then r, is surjective and Kerr*=

~ Cd~r.

Proof. First of all we see clearly that r* maps ~ ( V ~ V*)G into ~(V*)r~ and that r,(Jww) ~ y,n[. We need the following (14.3) Lemma. I f Ywc=~(V)vanishes on (9, then Y* has no H*-invariant other than O. Given this lemma, we can see the proof of Proposition 14.2 as follows. Suppose we have an element of ~(V*)n$ which does not come from ~ ( V ~ V*)~ under r*. Then we can find an irreducible component Y* ~_~(V*) in which there exists an H*-invariant linearly independent of re(J~). By Lemma 14.1, the dimension of H*-invariants in Y* is at most 1 (see the proof of Proposition 8.3). Therefore the situation above can happen only when r*(J~)=O and dimY*n~=l. But r*(J~w)= 0 means that a basis {Yi} in Yr vanishes simultaneously at v = v~, thus vanishes on the whole orbit (9. In other words, r*(Jw~)= 0 is equivalent to Yrl~= 0. So Lemma 14.3 gives us a contradiction. This also proves the assertion about the kernel of r*.

Proof of Lemma 14.3. Consider the restriction map Q : ~ ( V ) ~ C [ ~ ] . Clearly KerQ= ~

Yw.

re~ s

For ~v~ (9• assume Y* has a non-zero H*-invariant ~0. By Frobenius reciprocity, Yr is realized as a subrepresentation of the space of regular functions C[(9] on the orbit. (More explicitly, consider ~s(g)= for f ~ Yr- Then ~ f is a regular function on G invariant from the right under H , , which pushes down to define a regular function, still denoted ~y, on (9 ~ G/H*.) Since ~is affine and (9 is open in ~, we can find a non-zero regular function ~ ~ C[O] such that ~ . ~ f is regular on for all f ~ Yr- In fact, clearly C[(9] is contained in the field of rational functions on (9, which coincides with the field of fractions of C[(9], because ~ is affine ([Htn, Theorem 3.2]). Since Yr is finite dimensional, our assertion follows from this. Now consider any B-eigenvector Z in the B-module generated by ~ . ~ ( f c Y~). Then Z has its highest weight belonging to (9~, because (91 is upward hereditary. Since ~(V) is multiplicity-free, this gives us Z=O, a contradiction. Using the isomorphism ~(V) ~ -~ ~(V@ V*)~, we obtain a surjective homomorphism ~ : ~ ( V ) ~ ( V * ) n ~ with K e r ~ =(Qr)r~• Therefore (14.4) Corollary. ~(V*)nb is a polynomial ring isomorphic to C[Q~; ~ ~ P~ +(v), ~ r

(9•

Summary 15. Here we collect the information about multiplicity-free actions in the table below to see easily what we got. We do not, however, give it in full detail because the space is limited. For more details, especially for the explicit formulas for Capelli problems, see the sections indicated.

0

O.|

No

SpznOGL3

No

0

E6|

Yes

Spinlo|

Yes

No

Spin9|

G2|

Yes

Spin7|

(m=>4)

Sp,t|

Yes

Yes

Sp2,| (,,>-2)

(n_>_3)

Yes

Yes

SP2,|

0

0

A2GLm

Yes

o

Yes

0

1,2,3

(3)

o

(2)

(3)

(2)

1,2

(2)

1,2

• 9

1,2; 2

(3)

(2)

1,2

(6)

1,2,3,4;2,4

(6)

1,2;2 (3) 1,2,3;2,3,4

o

11.12.1

11.9.6

11'.8

11.6

(2)

(2)

o

(4)

m=4

•

(3)

o

1

(1)

•

1,2

1, 2..... En/2] (En/2])

(n)

1,2 ..... n

(m)

1,2~ ...~m

Degrees of fundamental generators (Number)

(2)

11.4.12

11.3.19

11.2.6

11.1.9

Explicit Capelli for rel.inv.

(2)

o

(n/2)

n: even

(.)

n-----m

(~

(Degree)

Relative invariant

Yes

Abstract Capr

0

Hermitian symmetric

GLn| GLm (n> m) S2GL,,

Group action

(15.1) Table

2,6; 1 2, 5,6, 8,9,12; 1

2, 4, 6; 1 2,4,6,8; 1 2, 4, 6, 8,10; 1

2,4, ...,2n; 1 2,4 ..... 2n; 1,2 2,4 ..... 2n; 1,2,3 2, 4; 1,2 ..... m

2, 4 ..... 2En/2] 1

1,2, ...,n

1, 2,..., n; 1,2, ...,m 1,2 .... ,n

Degrees of generators of . ~ ( g )

St,.

(3)

Linear

(2)

Linear

(2)

Linear

(3)

Linear

(2)

Linear

F4

SL3

Spin7

G2

Sp4 if m = 4

Linear

(13.5)

GL2

On- 1

if n even

(5)

(5)

Not linear:

(3)

Linear

(1)

Linear

Linear (2)

(I-n/2])

O.

GI_.~ ff n=m

Linear

Genetic stabilizer ffreducfive

(m) Linear (n) Linear

Graph of closure relations (Number of singular orbits)

H

r

g

o

t~

T h e Capelli i d e n t i t y

613

Appendix Some central elements in '~/(gl.) and ~(a.)

A.1. Case gI~.We continue the discussion in Sect9 11.1 just after (I 1.1.13). We want to show C,(z) is central in q/(gI~). First of all, we can easily see that C,(z) commutes with the diagonal elements E , (i= 1..... n). In fact, in the definition of the determinant, each monomial like EI~O)...Ei~(i)...E~- qoi. . .End(.) commutes with E u, because [ E i i , Ei~(O] = Eio~o and [ E i t , Ea-1(i)i3 " ~ - - - Eo- l ( i ) l yield the opposite signatures. Since E,+ ~ and E~+ ~i (i = 1.... , n - 1) generates gI, as a Lie algebra, we have only to prove that C,(z) commutes with those elements. In general, let us consider what conditions should be posed on the numbers (z~, ..., z,) if det(E+diag(z~ ..... z,)) is in ~q/(gl,). Here E=(Etj)Tj=I. If z i = n - i - z , this determinant is just C,(z). For simplicity we set E(z)=E + diag(zl ..... z~). Define i~ as a matrix whose entries are given by the commutator o f E , +~ and the elements of E (or the same thing replaced with E(z)). Explicitly we have E = E, + u --E ~ where 0 ...0...0

-0

:

:

..

:

0 ...0...0

0 (A.I.1)

..

E(~+ 1)= E l l

E l 2 ... E i i ... E i .

0

i+1,

0 ...0...0

i'..i".i 0 ...0. 9

0 and

(A.1.2)

E u) =

0 0

... ...

0 0

E1~+1 E2i+l

0 0

... ...

0 0

:

...

"

:

:

-..

."

0 :

... ..9

0 :

Ei+li+l :

0 :

... ..

0 ,

0

...

0

E,i+l

0

...

0

In (A.I.1), the notation (i+ 1) points out the (i+ 1)*t row as the row of interest. Similarly in (A. 1.2), the _/points out the ith column. The same notation is used in (A.1.4) below. We can also write this as 1~= l~i+ 1)- I~~ where l~ti+ ~)is the matrix (0 is the matrix E"0) except that E{~+~) except that Eii is replaced with E, + z~ and I~" Ei+~+~ is replaced with E~+~i+~+z~ respectively. We shall compute the commutator D = [E, + 1, detE(z)]. Using the multi-linearity of the determinant, we see D = ~ Dr - D'i, where j=l

614

R. Howe and T. Umeda

(1) Dj is the determinant of the matrix E(z) obtained by replacingj th column of E(z) by the jt~ column of Eti+ 1), (2) D'~is the determinant of the matrix F4z) obtained by replacing ith column of E(z) by the jth column of Et0. Observe that (1') Djis also equal to the determinant of the matrix E(z) obtained by replacing (i + 1)th row of E(z) by the following row vector whose entries are zeros other than jtla component

Ei+li+~+zi

[0... 0

0 . . . 0].

We can see (1') from the following simple trick: 0 0 (A.1.3)

det

...

9

a

9

...

= det

... 0

a

0...0

0 0 After this replacement in D j, collect the sum ~ D~. Then we obtain a determinant i=1

expression of a matrix with two identical rows (ith and (i + l)th), which determinant vanishes. Thus, what is remaining is the term D'~, which reads:

(A.1.4)

D~=det

*

Eli+I

Eli+l

*

*

E2i+l

E2i+l

*

*

Eli+ 1

Eli+ I

*

*

* *

Ei+li+l-+-2

Ei+2i+t Eni+ 1

i

Ei+li+l+Zi+l

*

Ei+2i+l

*

Eni+ 1

*

Expanding this determinant with respect to the adjacent two columns, ith and (i+ 1)th, by Laplace theorem, we can see D'~= 0 ifall 2 x 2 minors contained in those two/th and (i+ 1)tb columns vanish9 This can be certified as follows. First, observe Eji+l and E~+ 1 commute with each other if neither j nor k coincides with i + 1. Next we have

Eji+l EJt+l ]=(zi+l--zi)Eji+l +[Ejt+I,E~+II+I] I_E~+tl+t+zt Ei+li+l+zi+l

det [ (A. 1.5)

= (zi + 1- z~+ 1)Et~+ 1.

The Capelli identity

615

Therefore D ~ = 0 if and only if z i + l = z ~ - l . This gives us the conclusion 9 The c o m m u t a t o r with E~+ 1~ is quite similar to that of Eu+ 1. Remarks 9 (a) If we replace E with its transposed one E t, then we should put the diagonal element (0 ..... n - 1) instead of ( n - 1,..., 0). (2) There exits a central element in q/(o,) expressed as a determinant 9 We can show this fact by a similar but a bit more complicated a r g u m e n t below 9 But we do not have any similar expression for the symplectic Lie algebras 9

A.2. Case o,. Let us take a usual realization of o, as

o.={X~gl.; x+x'=o}, so that the standard basis consist of A ~ i = E i i - E j i (i<j). Define A = ( A 0 7 . j = 1. Similarly for (A.1), we raise a question whether det(A + diag(z 1..... z.)) belongs to ~~ for a suitable (zl ..... z.). Since Au+ 1(i = 1 ..... n - 1 ) generates o. as a Lie algebra, we have only to check [Aii+ 1, det(A + diag(zl ..... z.))] = 0. F o r simplicity we put A(z) = A + diag(zl ..... z.). Define ~ as a matrix whose entries are given by the c o m m u t a t o r of A , + 1 and the elements of A (or the same thing replaced with A(z)). Explicitly we have ~ = A(~+ 1) - A(~)+ A (i + 1)_ A(/), where for I = 1, i + 1 0

-0

... 0 ... 0

i 9149 . 9 0 ... 0 ... 0

0

(A.2.1)

A(z) =

0...0...0

0

9

, 9 .

A (1)=

.

9 ,,

0

0 ...0...0

-0 ... 0

Ali+l

0 ... 0-

0 ... 0

A2i+ 1

0 ... 0

9

(A9149

Ai ~ (t,

A i 2 ... A i i ...

All

,9

9

0...0 9 9

9

9

9

A~+I~+ 1 0 . . . 0

9149 9

.9

9149 .

0 ... 0

,

A,,+x

9

.

0 ... 0

Here (1 and I are the same convention as in Sect. A.I. We can rewrite this as J~ = -~(~+ 1 ) - -~(ij + ~(i + 1) _ ~(i), where "~(0 is the matrix A(i+ 1) except that At+ 1~+ 1 is replaced with At+ 1~+ 1 + z~+ 1, - ~ + 1) is the matrix A(i + 1) except that A~i is replaced with A~ + z~, ~(0 is the matrix A (~ except that Ai + 1i+ 1 is replaced with A i +1~+ 1 + z~, and j~(~+ i) is the matrix A" + 1) except that A , is replaced with Au + zi + 1 respectively. Let us c o m p u t e the c o m m u t a t o r D = [Au+ 1, detA(z)]. Using the multilinearity of the determinant, we see D = ~ (D(i + 1)j- D(oi) + D(i + 1) _ D(O, j=l

616

R. Howe and T. Umeda

where (1) for k = 1, i+ 1, D~k)j is the determinant of the matrix A(z) obtained by replacing jib column of A(z) by the jth column of ~[~k), (2) for k = i, i + 1, D ~k)is the determinant of the matrix A(z) obtained by replacing kth column of A(z) by the kth column of ~(k). We can see by the same trick (A.1.3) as the gl, case that (1') for k = i, i + 1, D(k)j is also equal to the determinant of the matrix A(z) 9 obtained by replacing ktli row of A(z) by the following row vector whose entries are zeros other than flh component

[0...0

z~, 0 . . . 0 3 ,

where k' = i + 1 o r i respectively for k = i o r i + 1. Summing up ~ D~k)jfor k = i, i + 1, j=l

we get two determinants of matrices whose itb and (i + 1)th rows are identical. Thus these terms vanish. N o w what is remaining are the terms D (~+1 ) - D ~~ which look like

(A.2.3)

Ali+,

All+ 1

,"

A2i + 1

ax/+ I

*

A.+ 1

Aii+ 1

zi

Zi+l

,

A/+2i+ 1

ai+2i+l

*

a._li+l

*

D") = det 9

9a n - l i +

1

Am + l

(A.2.4)

D0+ ~)=det

.

*

Ani + 1

*

Ali

AI~

*

A2i

A2~

*

Ai-li

a i - ti

*

Zi

Zi+ 1

.

A i + li

A i + li

*

A n - 1~

a._ li

*

An~

Ani

*

Ii+1,

(i .

Expanding those determinants with respect to two columns, i th and (i+ l) lh, by Laplace theorem, we can see D "+ 1)= D u) it'~lfthe 2 x 2 minors contained in these two columns are the same. This can be checked as follows. First for k, 14= i, i + 1, the 2 • 2 minors, which amount to be the commutators, coincide: [Aki + 1, Au + 1] --- - Akt = [Aki, Au'l

The CapeUi identity

617

N e x t for k ~: i, c o m p a r e the 2 • 2-minors from k th and ( i + I) st rows: det [At,~+ t

Ak,+,]z,+IA=(Zi+l

_zi)Akt+ 1

Aki Aki ]=[Ak~,Ai+lj=_Ak~+ L&+xi Ai+li_l 1"

detF

Therefore these coincide if and only if z i+ 1 - z i = - 1. An easy check for k th and ith rows gives us the same condition as above. A n d the last check for the 2 x 2 minors in ith and ( i + 1 ) st rows turns out a trivial condition A u + l = - A i + t i , which automatically holds. Thus we conclude det A(z) ~ ~q/(o~) if and only ifz~+ 1 = z ~ - 1 for i = 1 , . . . , n - 1.

Remarks.

(1) Similarly as in gI~ case, if we replace A with its transpose A t, the condition on (zl ..... z~) should be replaced by z i + ~= z~ + 1 instead of z~§ 1 = z i - 1. (2) If n is odd, detA(z) is an element of at most degree n - 1 . In this case, if we n+l specialize z~ as - - - f - - i , then this determinant vanishes. (3) The expression above depends heavily on the realization of o~. If we realize the o r t h o g o n a l Lie algebra using a general symmetric matrix S as

o.(s) = { x e gl.; x s + s x ' = 0}, we cannot expect the same p r o o f to be valid.

Acknowledgements.The second author would like to express his sincere thanks to Yale University and Univerist6 Paris VII for their hospitality during his stay. References [Ba] [Bo] EBI] EB2] FBW] [Br]

[Call [Ca2] [Ca3] [CL] FD]

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618 [FZ]

[G] [Ha] [Htn] [Hel] [He2] [He3] [HI] [S2] In3] [H4] [S5] [H6] [Hu] [I] [Ja] [Jb] [J] [K] [KPV] [KT]

[KS] [Kz] [Kr] ILl [Mc] [MRS] [Mlq IRa]

JR] [RS] IS]

[SK]

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The Capelli identity [Se] [Schl] [Sch2]

EScb3-1 [Sch4] [Sch5] [Sch6]

EShl] ESh2] ETh-I [TI] EX2] [VK]

EW] [zs] EZb] [Z]

619

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