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 ~a,

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

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

if b 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 _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 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 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'~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]

Bailey, W.N.: Generalized hypergeometric series. Cambridge Math. Tracts 32 (repr. Harrier 1965). Cambridge: Cambridge Univ. Press 1935. Boole, G.: Calculus of finite differences. New York: Chelsea 1872 Borel, A.: Hermann Weyl and Lie Groups, in "Hermann Weyl 1885-1985," (Centenary Lectures) K. Chandrasekharan (ed.), pp. 53-82. Berlin Heidelberg New York: Springer 1986 Borel, A.: Linear algebraic groups. New York: Benjamin 1969 Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representation of reductive groups. Princeton: Princeton Univ. Press 1980 Brion, M.: Classification des espaces homog~nes sphrriques. Compos. Math. 63,189-208 (1987) Capelli, A.: ~ber die Zuriickfiihrung der Cayley'schen Operation f2 auf gewfhnlichen Polar-Operationen. Math. Ann. 29, 331-338 (1887) Capelli, A.: Ricerca delle operazioni invariantive fra piu serie di variabili permutabili conogni altra operazione invariantive fra le stesse serie. Atti Sci. Fis. Mast. Napoli (2) I 1-17 (1888) Capelli, A.: Sur les oprrations darts la throrie des formes algrbriques. Math. Ann. 37, 1-37 (1890) Carter, R.W., Lusztig, G.: On the modular representations of the general linear and symmetric groups. Math. Z. 136, 193-242 (1974) Dixmier, J.: Sur les algrbres enveloppants de ~I(2,C) et affn, C), Bull. Sci. Math. I1 Ser. 100 57-95 (1976)

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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