Quadratic Capelli operators and Okounkov polynomials
aa r X i v : . [ m a t h . R T ] J a n QUADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS
SIDDHARTHA SAHI AND HADI SALMASIAN
Abstract.
Let Z be the symmetric cone of r × r positive definite Hermitian matrices over a realdivision algebra F . Then Z admits a natural family of invariant differential operators – the Capellioperators C λ – indexed by partitions λ of length at most r , whose eigenvalues are specializations ofKnop–Sahi interpolation polynomials.In this paper we consider a double fibration Y ←− X −→ Z where Y is the Grassmannian of r -dimensional subspaces of F n with n ≥ r . Using this we construct a family of invariant differentialoperators D λ,s on Y that we refer to as quadratic Capelli operators. Our main result shows that theeigenvalues of the D λ,s are specializations of Okounkov interpolation polynomials. Keywords:
Grassmannian manifolds, Harish-Chandra homomorphism, Okounkov polynomials, qua-dratic Capelli operators, symmetric cones.
MSC 2010 : 05E05, 22E46
R´esum´e . Soit Z le cˆone sym´etrique de matrices de tailles r × r hermitiennes positives sur une alg`ebrede division r´eelle F . Alors Z admet une famille naturelle d’op´erateurs diff´erentiels invariants - les Op´erateurs de Capelli C λ - index´es par des partitions λ de longueur au plus r , dont les valeurspropres sont des sp´ecialisations de polynˆomes d’interpolation Knop-Sahi.Dans cet article, nous consid´erons une double fibration Y ←− X −→ Z o`u Y est la vari´et´egrassmannienne des sous-espaces de dimension r de F n avec n ≥ r . En utilisant cela, nous con-struisons une famille d’op´erateurs diff´erentiels invariants D λ,s sur Y que nous appelons op´erateursCapelli quadratiques . Notre r´esultat principal montre que les valeurs propres des D λ,s sont dessp´ecialisations de polynˆomes d’interpolation Okounkov. Mot cl´e:
Vari´et´es grassmanniennes, homomorphisme de Harish-Chandra, polynˆomes Okounkov,op´erateurs Capelli quadratiques, cˆones sym´etriques. Introduction
Let F = R , C , H be a real division algebra. Fix integers r and n such that 1 ≤ r ≤ n . Let Y be the Grassmannian of r -dimensional subspaces of F n , and let Z be the symmetric cone of r × r positive definite Hermitian F -matrices. Then one has a double fibration X ϕ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ψ ❅❅❅❅❅❅❅❅ Y Z where X is the space of n × r matrices of F -rank r . For x ∈ X , ϕ ( x ) is the column space (or range)of x , while ψ ( x ) := x † x , where x † denotes the F -Hermitian adjoint of x .One can give another description of the above structure in terms of the groups G m := GL m ( F ) , K m := U m ( F ) := n g ∈ G m : g † g = I m × m o . The groups K n and G r act on X by matrix multiplication on the left and right respectively, andthe maps X ψ −→ Z and X ϕ −→ Y are simply the corresponding quotient maps. Moreover, X ϕ −→ Y is a principal G r -bundle, while X ψ −→ Z is a fibration whose fibers are isomorphic to the Stiefelmanifold K n /K n − r . Also, since the actions of K n and G r on X commute, it follows that G r acts on Z , and K n acts on Y . In fact, Y and Z are symmetric spaces for the latter actions. More precisely,we have Y ≃ K n / ( K r × K n − r ) and Z ≃ G r /K r . The cone Z is a symmetric space of type A , and admits an important basis of G r -invariantdifferential operators C λ , indexed by partitions λ ∈ P r , where P r := { ( λ , . . . , λ r ) ∈ Z r : λ ≥ · · · ≥ λ r ≥ } . The operators C λ were first studied by the first author in [20], and were referred to as Capellioperators . It is known that the spectrum of C λ is given by specialization of Knop–Sahi type A interpolation polynomials [20], [12], [30].On the other hand, Y is a compact symmetric space of type BC . In this paper, we use theabove double fibration to construct a family of K n -invariant differential operators D λ,s on Y thatcorrespond to the Capelli operators C λ . We call the operators D λ,s the quadratic Capelli operators because they are obtained from C λ by pullback of the quadratic map ψ . Our main result proves thatthe spectrum of D λ,s is given by specialization of the Okounkov type BC interpolation polynomials P λ ( x ; τ, α ) (see [14, Sec. 5.3] and [19]).To describe our main result precisely, we begin by introducing some notation. Set K := K n and M := K r × K n − r ⊂ K , so that Y ≃ K/M . The group K acts by left translation on C ∞ ( Y ), the spaceof complex-valued smooth functions on Y . The operators D λ,s leave the subspace C ∞ ( Y ) K -finite of K -finite vectors invariant. By standard results from the theory of compact symmetric spaces (forexample, see [5, Chap. V]), C ∞ ( Y ) K -finite decomposes as a multiplicity-free direct sum of irreducible M -spherical K -modules, which are naturally parametrized by partitions µ ∈ P r . Our next goal isto describe this parametrization. Let k and m denote the Lie algebras of K and M . Fix a Cartandecomposition k = m ⊕ p . Let a ⊆ p be a Cartan subspace, and let h be a Cartan subalgebra of k such that a ⊆ h . Then h = t ⊕ a , where t := h ∩ m . Set k C := k ⊗ R C , h C := h ⊗ R C , a C := a ⊗ R C ,and t C := t ⊗ R C . The restricted root system Σ := Σ( k C , a C ) is of type BC r . We choose a positivesystem Σ + ⊂ Σ and a basis e , . . . , e r of a ∗ C such that the multiplicity m α of every α ∈ Σ + is givenin terms of n , r , and d := dim F in Table 1 below. α e i , 1 ≤ i ≤ r e i ± e j , 1 ≤ i < j ≤ r e i , 1 ≤ i ≤ rm α d ( n − r ) d d − k C , h C ) which is compatible with Σ + .Let e µ ∈ h ∗ C . By the Cartan–Helgason Theorem, e µ is the highest weight of an irreducible M -spherical K -module if and only if(1) e µ (cid:12)(cid:12) t C = 0 and e µ (cid:12)(cid:12) a C = r X i =1 µ i e i , where µ := ( µ , . . . , µ r ) ∈ P r . Remark 1.1.
Assume that F = R . Then K is disconnected, and if K ◦ denotes the connectedcomponent of identity of K , then M ∩ K ◦ is also disconnected. Therefore the Cartan–Helgason UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 3
Theorem as stated for instance in [5, Cor. V.4.2] does not apply immediately to the case F = R .However, one can use the refinement of the Cartan–Helgason Theorem for the pair ( K ◦ , M ∩ K ◦ ),given in [4, Sec. 12.3.2], as well as the description of irreducible representations of K in terms ofirreducible representations of K ◦ , given in [4, Thm 5.5.23], to obtain the condition (1).From now on, we denote the M -spherical K -module with highest weight e µ satisfying (1) by V µ .Therefore as K -modules, C ∞ ( Y ) K -finite ≃ M µ ∈P r V µ . The operator D λ,s acts on V µ by the scalar c λ,s ( µ ) := HC( D λ,s ) (cid:0)e µ | a C + ρ (cid:1) , where ρ := P α ∈ Σ + α , e µ is the highest weight of V µ , and HC : D K ( Y ) → P ( a ∗ C ) W is the Harish-Chandra homomorphism from the algebra D K ( Y ) of K -invariant differential operators on Y ontothe algebra of polynomials on a ∗ C that are invariant under the action of the restricted Weyl group W .We now recall the definition of the Okounkov polynomials P λ ( x ; τ, α ). Let k := C ( τ, α ) denotethe field of rational functions in τ and α . Let δ, ∈ P r and ̺ τ,α ∈ k r be defined by(2) δ := ( r − , . . . , , := (1 , . . . , , ̺ τ,α := τ δ + α . For λ ∈ P r , we define | λ | := P ri =1 λ i . Up to a scalar, P λ ( x ; τ, α ) ∈ k [ x , . . . , x r ] is the uniquepolynomial of degree 2 | λ | which is invariant under permutations and sign changes of x , . . . , x r , andsatisfies P λ ( µ + ̺ τ,α ; τ, α ) = 0for every µ ∈ P r such that | µ | ≤ | λ | and µ = λ (for more details, see Section 4).Recall that d := dim F . Let i : C r → a ∗ C be the linear map defined by i ( e i ) := 2 e i for 1 ≤ i ≤ r ,where e , . . . , e r are the standard basis vectors of C r (therefore i ( P r ) is the set of restrictions to a C of highest weights of M -spherical K -modules). Set ̺ := i − ( ρ ) . A simple calculation yields(3) ̺ = ( ̺ , . . . , ̺ r ) where ̺ i := dn − − d ( i − ≤ i ≤ r. We are now ready to state our main theorem.
Theorem 1.2.
For every λ, µ ∈ P r and every s ∈ C , the operator D λ,s acts on V µ by the scalar (4) c λ,s ( µ ) := γ λ P λ (cid:18) µ + ̺ ; d , s − ̺ (cid:19) , where γ λ is a certain explicit constant, defined in (18) below. We now briefly sketch the strategy behind the proof of Theorem 1.2. We first show that the equal-ity (4) holds up to a scalar multiple. In view of the characterization of the Okounkov polynomials P λ , it suffices to prove that c λ,s ( x − ̺ ) is a polynomial in x and s which has the same degree andvanishing property as P λ ( x ; τ, α ) for τ := d and α := s − ̺ (see Proposition 4.3 and Proposition5.6). It will be seen that verifying the pertinent vanishing property can be reduced to a slightlyweaker one, that is, to show that the operators D λ := D λ, vanish on certain V µ ⊂ C ∞ ( Y ) K -finite .Up to this point, the strategy is the same as the one in the case of Knop–Sahi type A polynomials. SIDDHARTHA SAHI AND HADI SALMASIAN
However, the proof of the vanishing property of the operators D λ is more subtle than the type A case, in that it does not follow from a direct reasoning that is based on orders of differential opera-tors. In addition, the ρ -shift of the symmetric space Y and ̺ τ,α are not identical. Rather, they arerelated to each other as in Remark 5.5.To overcome these difficulties, we need to use the fact that c λ,s ( x − ̺ ) is symmetric under permu-tations and sign changes of its variables, and therefore c λ,s ( x − ̺ ) = 0 for x := ( x , . . . , x r ) if andonly if c λ,s (˜ x − ̺ ) = 0 for ˜ x := ( − x r , . . . , − x ). The latter observation results in an equivalent formof the vanishing condition for c λ,s , which is verified in the proof of Proposition 5.6 using differentialoperator techniques and branching rules. The branching rule from G n to K n that we need in theproof of the vanishing property of D λ is the Littlewood–Richardson Rule for F = C and the Little-wood Restriction Theorem for F = R , H . As a result, some of our proofs are divided into two cases,but they lead to uniform statements.The last step in the proof of our main result is to determine the scalar γ λ that relates the twosides of (4). To this end, we use the fact that the top degree homogeneous term of P λ ( x ; τ, α ) is aJack polynomial. Using a trick which relies on an identity for Jack polynomials (see [28, Prop. 2.3]),calculation of γ λ for general λ is reduced to when λ corresponds to the 1 × G n and K n . This is carried out in the Appendix.We now describe the relation between Theorem 1.2 and earlier results on invariant differentialoperators on symmetric spaces. The Capelli operators C λ were originally studied in connection tothe famous Capelli identity, which has also been considerably generalized by Howe and Umeda [9]from the viewpoint of multiplicity-free actions, and by Kostant and the first author [15], [16] fromthe viewpoint of Jordan algebras. For F = R and λ = , the operator D , − was first considered byHowe and Lee in [7], who computed its spectrum for r = 2 and asked for the determination of thespectrum for general r . This was solved by the first author more generally for D m ,s and F = R ,[22], and subsequently by Zhang and the first author [25] for arbitrary F , where a connection withthe Radon transform was also established. The explicit form of the answer in [25] gave us the firsthint that the general situation might have something to do with the Okounkov polynomials. Theresult of [25] on the spectrum of D m ,s is indeed a special case of our Theorem 1.2.Finally, we say a few words about prospects for future research that emerge from this work.Quite recently, Zhang and the first author established another link between Okounkov interpolationpolynomials and the spectrum of Shimura operators on Hermitian symmetric spaces [26]. It wouldbe interesting to understand the connection between our main result and the results of [26]. Also,in view of our recent work [23] on the Capelli eigenvalue problem in the case of the supersymmetricpairs ( gl ( m | m ′ ) × gl ( m | m ′ ) , gl ( m | m ′ )) and ( gl ( m | m ′ ) , osp ( m | m ′ )), it is natural to ask whetherTheorem 1.2 can also be extended to the setting of Lie superalgebras. This is likely to involvethe deformed BC interpolation polynomials of [27]. We remark that in forthcoming papers [1] and[24], we extend the results of [23] to the setting of multiplicity-free actions obtained from Jordansuperalgebras. Another interesting problem is to extend the Littlewood Restriction Theorem (seeProposition 2.12) to the super setting, namely to ( gl ( m | m ′ ) , osp ( m | m ′ )). We are planning to studythese problems in the near future. Acknowledgement.
The authors thank Kyo Nishiyama and Nolan Wallach for helpful e-mailcorrespondences. The research of Siddhartha Sahi was partially supported by a Simons Foundationgrant (509766) and of Hadi Salmasian by an NSERC Discovery Grant (RGPIN-2013-355464). Partof this work was carried out during the Workshop on Hecke Algebras and Lie Theory held at the
UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 5
University of Ottawa during May 12–15, 2016. The authors thank the National Science Foundation(DMS-162350), the Fields Institute, and the University of Ottawa for funding this workshop.2.
Parametrization of representations by partitions
In this article, we will need various parametrizations of finite dimensional representations of G n , K n , and G r by partitions. Instead of working with representations of these real Lie groups, it willbe more convenient to work with representations of their complexifications.Let W := Mat n × r ( F ) denote the space of n × r matrices with entries in F . Furthermore, set A := { x ∈ Mat r × r ( F ) : x † = x } . Then X ⊂ W and Z := { w † w : w ∈ X } ⊂ A are open. The G n × G r -action on X is the restrictionof the G n × G r -action on W given by( g , g ) · w := g wg − for ( g , g ) ∈ G n × G r , w ∈ W. The G r -action on Z is the restriction of the G r -action on A given by g · x := ( g † ) − xg − for g ∈ G r , x ∈ A. Let G n and G r denote the complexifications of the real Lie groups G n and G r . Similarly, let K n ⊂ G n denote the complexification of K n . Remark 2.1.
From now on, we need to fix an embedding of quaternionic matrices into complexmatrices of twice the size. For integers k, m ≥
1, let x := a + b j ∈ Mat k × m ( H ), where a, b ∈ Mat k × m ( C ). We set ˘ x := " a − bb a ∈ Mat k × m ( C ) . Remark 2.2.
The matrix realizations of the embeddings i n : G n ֒ → G n and i r : G r ֒ → G r are asfollows. If F = R , then G n ≃ GL n ( C ) and i n : GL n ( R ) → GL n ( C ) is the obvious map. If F = C ,then G n ≃ GL n ( C ) × GL n ( C ) and i n : GL n ( C ) → GL n ( C ) × GL n ( C ) is the map g (( g ∗ ) − , g ),where g ∗ := g T is the adjoint of g . If F = H , then G n ≃ GL n ( C ) and i n : GL n ( H ) → GL n ( C ) isthe map g ˘ g . The matrix realization of K n as a subgroup of G n is as follows. If F = R , then K n = { g ∈ GL n ( C ) : g T g = I n × n } . If F = C , then K n = { ( g, g ) : g ∈ GL n ( C ) } . Finally, if F = H ,then K n = { g ∈ GL n ( C ) : g T J n g = J n } , where(5) J n := " I n × n − I n × n . The definition of the embedding i r : G r → G r is similar to that of i n . To help the reader, wesummarize the information about G n and K n in Table 2 below. F G n K n Realization of K n in G n R GL n ( C ) O n ( C ) { g ∈ GL n ( C ) : g T g = I n × n } C GL n ( C ) × GL n ( C ) GL n ( C ) { ( g, g ) : g ∈ GL n ( C ) } H GL n ( C ) Sp n ( C ) { g ∈ GL n ( C ) : g T J n g = J n } Table 2.
SIDDHARTHA SAHI AND HADI SALMASIAN
Set W C := W ⊗ R C and A C := A ⊗ R C . The map X ψ −→ Z has a unique holomorphic extension W C ψ −→ A C . For an explicit description of A C , W C , and the map ψ : W C → A C , see Table 3 below. F A C W C ψ R Sym r × r ( C ) Mat n × r ( C ) x x T x C Mat r × r ( C ) Mat n × r ( C ) ⊕ Mat n × r ( C ) ( x , x ) x T x H Skew r × r ( C ) Mat n × r ( C ) x
7→ − x T J n x Table 3.In Table 3, Sym r × r ( C ) denotes the space of complex symmetric r × r matrices and Skew r × r ( C )denotes the space of complex skew symmetric 2 r × r matrices. Remark 2.3.
The matrix realizations of the maps A → A C and W → W C are as follows. For F = R and F = C , the map A ֒ → A C is the obvious embedding, and for F = H , it is the map a
7→ − J r a ,where J r is defined similar to (5). In fact the map A ֒ → A C is related to realization of EuclideanJordan algebras (see [2, Sec. VIII.5]). As for W ֒ → W C , it is the obvious embedding if F = R , themap w ( w, w ) if F = C , and the map w ˘ w if F = H .The action of G r on A extends uniquely to a holomorphic action of G r on A C . Similarly, theaction of G n × G r on W extends uniquely to a holomorphic action of G n × G r on W C . Theseholomorphic actions are explicitly described in Table 4 below. F G r (cid:9) A C ( G n × G r ) (cid:9) W C R S ( C r ) ∗ C n ⊗ ( C r ) ∗ C C r ⊗ ( C r ) ∗ (( C n ) ∗ ⊗ C r ) ⊕ ( C n ⊗ ( C r ) ∗ ) H Λ ( C r ) ∗ C n ⊗ ( C r ) ∗ Table 4.
Definition 2.4.
For every integer m ≥
1, let H m ⊂ GL m ( C ) denote the standard Cartan subgroupof diagonal matrices, and let B m ⊂ GL m ( C ) denote the standard Borel subgroup of upper triangularmatrices. Let ε , . . . , ε m denote the standard generators of the weight lattice of GL m ( C ). For every λ := ( λ , . . . , λ m ) ∈ Z m such that λ ≥ · · · ≥ λ m , we denote the GL m ( C )-module with B m -highestweight P mi =1 λ i ε i by M λ . Definition 2.5.
The standard Borel subgroup of G m will be denoted by B m . In cases F = R , F = C , and F = H , the group B m ⊂ G m equals B m , B m × B m , and B m . The standard Cartansubgroup of G m will be denoted by H m . Remark 2.6.
For every integer m ≥
1, we define P m := { ( λ , . . . , λ m ) ∈ Z m : λ ≥ · · · ≥ λ m ≥ } .From now on, we will denote the length (that is, the number of nonzero parts) of a partition λ ∈ P m by ℓ ( λ ). For two partitions λ ∈ P m and µ ∈ P k , where k, m ≥
1, we write λ ⊆ µ if and only if ℓ ( λ ) ≤ ℓ ( µ ) and λ i ≤ µ i for every 1 ≤ i ≤ ℓ ( λ ).Let P ( A ) and P ( W ) denote the C -algebras of polynomials on A C and W C . The canonical G r -action on P ( A ), given by g · f ( a ) := f ( g − · a ) for g ∈ G r , f ∈ P ( A ), and a ∈ A , extends uniquelyto a holomorphic G r -action on P ( A ). Similarly, the canonical G n × G r -action on P ( W ) extends UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 7 uniquely to a holomorphic G n × G r -action on P ( W ). The pullback(6) ψ ∗ : P ( A ) → P ( W ) , f f ◦ ψ is a G r -equivariant embedding of C -algebras. The image of ψ ∗ is precisely described by the FirstFundamental Theorem of invariant theory [4, Sec. 5.2.1]. In particular, ψ ∗ ( P ( A )) = P ( W ) K n .By classical invariant theory (for example see [4]), P ( A ) decomposes into a direct sum of irre-ducible G r -modules which are naturally parametrized by partitions λ ∈ P r . Thus,(7) P ( A ) ≃ M λ ∈P r F λ , where F λ is the irreducible G r -module corresponding to λ ∈ P r . In fact F λ ≃ M ∗ λ ⊗ M λ if F = C , and F λ ≃ M λ • if F = R , H , where λ • := (2 λ , . . . , λ r ) ∈ P r if F = R , and λ • := ( λ , λ , . . . , λ r , λ r ) ∈ P r if F = H .The map (6) is G r -equivariant, and therefore F λ occurs as a G r -submodule of P ( W ) for every λ ∈ P r . Therefore by the well known (GL n , GL r ) duality (see [6, Sec. 2.1] or [4, Sec. 5.6.2]), forevery λ ∈ P r there exists a unique irreducible G n -module E λ such that E λ ⊗ F λ occurs in P ( W )as a G n × G r -submodule. Remark 2.7.
Let P K n ( W ) denote the direct sum of irreducible K n -spherical G n -submodules of P ( W ). Then indeed P K n ( W ) ≃ M λ ∈P r E λ ⊗ F λ as G n × G r -modules. Remark 2.8.
For every integer l ≥
1, let S l × l be the l × l matrix with 1’s in ( i, l − i + 1)-entry forevery 1 ≤ i ≤ l , and with 0’s elsewhere. Consider g ◦ ∈ G n defined as follows. If F = R , then we set g ◦ := " i I l × l − i S l × l − i S l × l i I l × l for n = 2 l, and g ◦ := i I l × l l × − i S l × l × l × l − i S l × l l × i I l × l for n = 2 l + 1 . If F = C , then we set g ◦ equal to the identity element of G n . Finally, if F = H , then we set g ◦ := " S n × n I n × n . We remark that when F = R , H , the map K n → G n , g g ◦ gg − ◦ is the embedding O n ( C ) ֒ → GL n ( C )or Sp n ( C ) ֒ → GL n ( C ) that is given in [4, Sec. 2.1.2]. In particular, in all cases ( g − ◦ H n g ◦ ) ∩ K n and ( g − ◦ B n g ◦ ) ∩ K n are Cartan and Borel subalgebras of K n .Set K := K n , and let M ⊂ K denote the complexification of M . The M -spherical K -module V µ is naturally also an M -spherical K -module. By comparing the calculation of highest weightsof M -spherical K -modules in [4, Sec. 12.3.2] (the pertinent cases are types BDI, AIII, and CII)with the parametrization of representations of K by partitions that uses generalized Schur–Weylduality (see [4, Thm 10.2.9] and [4, Thm 10.2.12]), it follows that for every µ ∈ P r , the module V µ is isomorphic to the K -submodule of E ∗ µ generated by g − ◦ · v λ , where v λ denotes the B n -highestweight of E ∗ µ . Definition 2.9.
Let G be a group, and let E and F be G -modules. We set[ E : F ] G := dim Hom G ( E, F ) . SIDDHARTHA SAHI AND HADI SALMASIAN
Remark 2.10.
In the following, we will need the Littlewood–Richardson Rule, which we now recall(for a more elaborate reference, see for example [18, Sec. I.9]). For a semistandard skew tableau T , the word w ( T ) corresponding to T is defined as the sequence of integers obtained by readingthe contents of boxes of T from right to left and from top to bottom. A word w · · · w k in letters { , . . . , N } is called a lattice permutation word if for every 1 ≤ i ≤ k and every 1 ≤ j ≤ N − j in w . . . w i is greater than or equal to the number of occurrencesof j + 1. Now let λ, µ, ν ∈ P m , where m ≥
1. The Littlewood–Richardson Rule states that[ M ν : M λ ⊗ M µ ] GL m ( C ) is equal to the number of tableaux T of shape ν \ µ and weight λ such that w ( T ) is a lattice permutation word. In particular, if [ M ν : M λ ⊗ M µ ] GL m ( C ) = 0, then µ, λ ⊆ ν . Lemma 2.11.
Let λ, µ ∈ P r . Then [ V µ : E λ ] K n = [ V µ : E ∗ λ ] K n .Proof. It is enough to show that E λ ≃ E ∗ λ as K n -modules. Let θ n : G n → G n denote the automor-phism of G n that is obtained by holomorphic extension of the Cartan involution g ( g † ) − of G n .Let E θ n λ be the G n -module that results from twisting E λ by θ n . Then E θ n λ ≃ E ∗ λ as G n -modules.Moreover, since θ n fixes K n pointwise, E λ ≃ E θ n λ as K n -modules. The K n -module isomorphism E λ ≃ E ∗ λ now follows immediately. (cid:3) When F = C , branching from G n to K n is described by the Littlewood–Richardson Rule. Thenext proposition is a branching from G n to K n when F = C . Proposition 2.12.
Assume that F = R or F = H , and let λ, µ ∈ P r . Then (8) [ V µ : E λ ] K n = X ξ ∈P r [ E λ : E µ ⊗ E ξ ] G n . Proof.
The statement follows as a special case of the Littlewood Restriction Theorem, which wasfirst proved in [17] (see also [13], [3], and [8, Sec. 1.3]). We now outline the calculations that areneeded to deduce the proposition from the Littlewood Restriction Theorem. We remark that the G n -modules that appear in the statement of the Littlewood Restriction Theorem are polynomialrepresentations, whereas the E λ are indeed contragredients of polynomial representations. To getaround this issue, we note that we can replace the left hand side of (8) by [ V µ , E ∗ λ ] K n (see Lemma2.11) and the terms of the right hand side by [ E ∗ λ : E ∗ µ ⊗ E ∗ ξ ] G n . Using Remark 2.8 we can writethe highest weight of V µ explicitly, and then we can determine the partition that corresponds to V µ in the parametrization of [4, Thm 10.2.9] and [4, Thm 10.2.12]. If [ E λ : E µ ⊗ M ξ ] G n = 0 for some ξ ∈ P n , then Remark 2.10 implies that ℓ ( ξ ) ≤ r when F = R , and ℓ ( ξ ) ≤ r when F = H . Byputting all of these facts together, we can verify that the statement of the proposition is a specialcase of the Littlewood Restriction Theorem. (cid:3) The quadratic Capelli operators
In this section we define the differential operators D λ,s . Let D ( A ) := L ∞ m =0 D m ( A ) denote the C -algebra of constant coefficient differential operators on A , endowed with the usual Z -grading. Wedefine D ( W ) := L ∞ m =0 D m ( W ) similarly. There are natural G r -actions on D ( A ) and D ( W ), and asin Section 2, these actions extend uniquely to holomorphic G r -actions on the same vector spaces.Furthermore, the canonical isomorphisms(9) D m ( A ) ≃ P m ( A ) ∗ and D m ( W ) ≃ P m ( W ) ∗ are G r -equivariant. UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 9
Let PD ( A ) and PD ( W ) denote the algebras of polynomial coefficient differential operators on A and W . The multiplication map results in isomorphisms of vector spaces(10) P ( A ) ⊗ D ( A ) ≃ PD ( A ) and P ( W ) ⊗ D ( W ) ≃ PD ( W ) . From now on, we set θ : G r → G r , θ ( g ) := ( g † ) − . Definition 3.1.
We define bilinear forms ( · , · ) W : W × W → R and ( · , · ) A : A × A → R by( x, y ) W := ℜ (tr( x † y )) and ( x, y ) A := ℜ (tr( xy )) . The bilinear form ( · , · ) W is K n -invariant and θ -invariant, that is,( kx, ky ) W = ( x, y ) W and ( g · x, y ) W = ( x, θ ( g ) − · y ) W for x, y ∈ W, k ∈ K n , and g ∈ G r . Similarly, the bilinear form ( · , · ) A is θ -invariant, that is, ( g · x, y ) A = ( x, θ ( g ) − · y ) A for x, y ∈ A and g ∈ G r . The bilinear forms ( · , · ) W and ( · , · ) A yield canonical isomorphisms(11) ι W : W → W ∗ and ι A : A → A ∗ . These maps extend to G r -equivariant isomorphisms of C -algebras(12) ι W : D ( W ) ≃ P ( W ) and ι A : D ( A ) ≃ P ( A ) . Since ι W is also K n -equivariant, it restricts to an isomorphism P ( W ) K n → D ( W ) K n . Consequently,we obtain a G r -equivariant isomorphism of C -algebras( ι W ) − ◦ ψ ∗ ◦ ι A : D ( A ) → D ( W ) K n . Set ˘ ψ := ( ι W ) − ◦ ψ ∗ ◦ ι A . From (10) it follows that the map(13) ι : PD ( A ) ֒ → PD ( W ) K n , ι := ˘ ψ ⊗ ψ ∗ is an embedding of G r -modules. From (7) and (9) it follows that D ( A ) ≃ L λ ∈P r F ∗ λ , so that PD ( A ) ≃ P ( A ) ⊗ D ( A ) ≃ M λ,µ ∈P r F λ ⊗ F ∗ µ ≃ M λ,µ ∈P r Hom C ( F µ , F λ ) . By Schur’s Lemma [ F µ : F λ ] G r ≤
1, and equality occurs if and only if λ = µ . Thus,(14) PD ( A ) G r ≃ M λ,µ ∈P r Hom G r ( F µ , F λ ) ≃ M λ ∈P r C I λ , where I λ denotes the identity element of Hom C ( F λ , F λ ). Let C λ ∈ PD ( A ) G r be the differentialoperator that corresponds to I λ by the isomorphism (14). We now set(15) e D λ := ι ( C λ ) ∈ PD ( W ) K n × G r , where ι is the map defined in (13).Set Ψ( x ) := det( x † x ) for x ∈ W (in the case F = H we define det( z ) := det(˘ z ) for z ∈ Mat r × r ( H ),where ˘ z is as in Remark 2.1). For every λ ∈ P r and every s ∈ C , let e D λ,s be the differential operatoron X defined by e D λ,s := Ψ s e D λ Ψ − s . Definition 3.2.
For λ ∈ P r and s ∈ C , we set ( D λ,s f )( ϕ ( x )) := e D λ,s ( f ◦ ϕ )( x ) for every f ∈ C ∞ ( Y )and every x ∈ X , where X ϕ −→ Y is the map defined in Section 1. Since D λ,s does not increase supports, by Peetre’s Theorem [5, Thm II.1.4] it is a differentialoperator on Y . From K n -invariance of e D λ,s it follows that D λ,s is also K n -invariant.4. The polynomials P λ ( x ; τ, α )In this section we review the definition and properties of the polynomials P λ ( x ; τ, α ). Definition 4.1.
Let λ := ( λ , . . . , λ r ) ∈ Z r . A Laurent polynomial f ( x , . . . , x r ) in variables x , . . . , x r is called λ -monic if the coefficient of x λ · · · x λ r r in f ( x , . . . , x r ) is equal to 1.Recall that the Weyl group W of type BC r is a semidirect product W := S r ⋉ {± } r , where S r denotes the symmetric group on r letters. In [19], Okounkov defined a family of Laurent polynomials P ip λ ( x ; q, t, a ) ∈ C ( q, t, a )[ x ± , . . . , x ± r ] , parametrized by partitions λ ∈ P r (we use the notation of [14, Sec. 5]). Every P ip λ is the unique λ -monic Laurent polynomial of degree | λ | that is invariant under the action of W on the x i ’s bypermutations and inversions, and satisfies the vanishing condition P ip λ ( aq µ t δ ; q, t, a ) = 0 unless λ ⊆ µ, where δ := ( r − , . . . ,
0) and µ ∈ P r (see [14, Sec. 5.3]). Here as usual we define q µ t δ :=( q µ t δ , . . . , q µ r t δ r ). The polynomials P ip λ ( x ; q, t, a ) are analogues (for the BC r -type root system)of the q -deformed interpolation Macdonald polynomials defined by Knop [11] and Sahi [21].By taking the q → P ip λ (see [14, Def. 7.1]), one obtains a polynomial P λ ( x ; τ, α ) ∈ C ( τ, α )[ x , . . . , x r ]. More precisely, P λ ( x ; τ, α ) := lim q ↑ (1 − q ) − | λ | P ip λ ( q x ; q, q τ , q α ) , where | λ | := P i λ i . From the symmetry property of P ip λ it follows that P λ ( x ; τ, α ) is invariant underpermutations and sign changes of x , . . . , x r . Definition 4.2.
A polynomial in variables x , . . . , x r is called even-symmetric if it is invariant underpermutations and sign changes of the x i ’s.A combinatorial formula for P λ ( x ; τ, α ) is given in [14, Sec. 7]. To recall this formula, we needsome terminology. Every partition λ can be represented by a Young diagram consisting of boxes ♭ := ♭ ( i, j ), where ( i, j ) ∈ { ( p, q ) ∈ Z : 1 ≤ p ≤ ℓ ( λ ) and 1 ≤ q ≤ λ i } . The arm length and leg length of a box ♭ := ♭ ( i, j ) in the Young diagram of λ are a λ ( ♭ ) := λ i − j and l λ ( ♭ ) := |{ k > i : λ k ≥ j }| . We also set a ′ λ ( ♭ ) := j − l ′ λ ( ♭ ) := i − λ with entries in { , . . . , r } we mean a filling ofthe Young diagram that corresponds to λ , with weakly decreasing rows and strongly decreasingcolumns. For a reverse tableau T of shape λ and an integer k ∈ { , . . . , r } , let λ ( k ) ⊆ λ be thepartition corresponding to the boxes ♭ ∈ λ that satisfy T ( ♭ ) > k . Thus for 1 ≤ k ≤ n , λ ( k − \ λ ( k ) is the horizontal strip consisting of the boxes that contain k . Finally, for two partitions ν ⊆ µ , wedefine ( R \ C ) µ \ ν to be the set of boxes which are in a row of µ intersecting with µ \ ν , but not in acolumn of µ intersecting with µ \ ν . Set b µ ( ♭ ; τ ) := a µ ( ♭ ) + τ ( l µ ( ♭ ) + 1) a µ ( ♭ ) + τ l µ ( ♭ ) + 1 and ψ T ( τ ) := r Y i =1 Y ♭ ∈ ( R \ C ) λ ( i − \ λ ( i ) b λ ( i ) ( ♭ ; τ ) b λ ( i − ( ♭ ; τ ) . UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 11
Then(16) P λ ( x ; τ, α ) = X T ψ T ( τ ) Y ♭ ∈ λ (cid:16) x T ( ♭ ) − ( a ′ λ ( ♭ ) + τ ( n − T ( ♭ ) − l ′ λ ( ♭ )) + α ) (cid:17) , where the sum is over all reverse tableaux T of shape λ with entries in { , . . . , r } . Proposition 4.3.
Fix real numbers α, τ > . For every partition λ ∈ P r , the polynomial Q λ ( x ) := P λ ( x ; τ, α ) ∈ C [ x , . . . , x r ] is the unique λ -monic, even-symmetric polynomial that satisfies the vanishing condition (17) Q λ ( µ + ̺ τ,α ) = 0 if | µ | ≤ | λ | and µ = λ, where ̺ τ,α is given in (2) .Proof. The existence statement follows from [14, Sec. 7] and the fact that the specialization of P λ at the values of τ and α is well-defined, because when τ >
0, the denominators of the coefficients ψ T ( τ ) of P λ ( x ; τ, α ) that appear in the combinatorial formula (16) do not vanish. For the uniquenessstatement, we use a method based on [20]. First note that from α, τ > Q λ ( λ + ̺ τ,α ) = 0. Next fix an integer N >
0, set I N := { µ ∈ P r : | µ | ≤ N } , and let S N denote the vector space of even-symmetric polynomials in x , . . . , x r of degree at most 2 N . Notethat dim S N = |I N | . For every µ ∈ I N , we consider the linear maps L µ : S N → C , f f ( µ + ̺ τ,α ) . Next we define a total order ≺ on I N , as follows. We set µ ≺ ν for every µ, ν ∈ I N which satisfy | µ | < | ν | , and then we extend the resulting partial order to a total order on I N . Then the matrix (cid:2) L µ ( Q µ ′ ) (cid:3) µ,µ ′ ∈I N is upper triangular with nonzero entries on the diagonal. It follows that the linear map S N → C dim S N , f (cid:2) f ( µ + ̺ τ,α ) (cid:3) µ ∈I N . is invertible. Uniqueness of Q λ follows immediately from the latter statement. (cid:3) Recall that d := dim( F ). For every λ ∈ P r , we set(18) γ λ := ( − d ) | λ | Q ♭ ∈ λ (cid:0) d ( a λ ( ♭ ) + 1) + l λ ( ♭ ) (cid:1) . This is the scalar that appears in the statement of Theorem 1.2.5.
The vanishing property of c λ,s ( µ )In this section, we prove a few technical statements which will be used in the proof of Theorem1.2. The ultimate goal of this section is to prove Proposition 5.6. The proof of Theorem 1.2 will becompleted in Section 6. Recall that ̺ := ( ̺ , . . . , ̺ r ) is the vector defined in (3). Lemma 5.1.
Let λ ∈ P r . Then there exists a polynomial d λ ( x, s ) which is even-symmetric in x := ( x , . . . , x r ) , has x -degree and s -degree at most | λ | , and satisfies c λ,s ( µ ) = d λ ( µ + ̺, s ) , for every s ∈ C and every µ ∈ P r . Proof.
The proof is similar to [25, Lemma 3.1]. Throughout the proof we fix λ . Recall that K := K n and M := K r × K n − r . For every s ∈ C , the differential operator D λ,s is K -invariant and has orderat most 2 | λ | . By (1), for every µ ∈ P r , the highest weight e µ of V µ satisfies e µ (cid:12)(cid:12) a C = P ri =1 µ i e i .Therefore from the Harish-Chandra homomorphism [5, Chap. II] it follows that for every s ∈ C ,the scalar c λ,s ( µ ) is an even-symmetric polynomial of degree at most 2 | λ | evaluated at µ + ̺ . Nextwe set x ◦ := " I r × r ∈ X, and we denote the image of x ◦ in Y by y ◦ . Choose an M -fixed vector h µ ∈ V µ ⊆ C ∞ ( Y ) such that h µ ( y ◦ ) = 1. By evaluating both sides of the relation D λ,s h µ = c λ,s ( µ ) h µ at y ◦ , we obtain(19) c λ,s ( µ ) = e D λ (det( x † x ) − s (cid:0) h µ ◦ ϕ ) (cid:1) ( x ◦ ) , where e D λ is defined in (15). Since e D λ is a polynomial coefficient differential operator of order atmost 2 | λ | , from the Leibniz rule it follows that for fixed µ , the right hand side of (19) is a polynomialin s of degree at most 2 | λ | . Consequently, c λ,s ( µ ) = | λ | X j =0 a j ( µ ) s j for every s ∈ C . For 2 | λ | + 1 distinct values of s , we obtain a linear system in the coefficients a j ( µ ) whose coefficientsform an invertible Vandermonde matrix. Since fox fixed s ∈ C we have shown that c s,λ ( µ ) is apolynomial in µ of degree at most 2 | λ | , it follows that a j ( µ ) is also a polynomial in µ of degree atmost 2 | λ | . Consequently, both the x -degree and the s -degree of c s,λ are at most 2 | λ | . The statementthat d λ is even-symmetric follows from the fact that c λ,s ( µ ) is even-symmetric as a polynomial in µ + ̺ . (cid:3) Definition 5.2.
For every χ ∈ P r , let P χ ( W ) denote the F χ -isotypic component of P ( W ).We remark that P χ ( W ) is G n -invariant. Proposition 5.3.
Let λ, ν ∈ P r , and let m ∈ Z such that m ≥ max { λ , ν } . Set η := ( m − ν r , . . . , m − ν ) and χ := ( m − λ r , . . . , m − λ ) . If [ V η : P χ ( W )] K n > , then λ ⊆ ν .Proof. Recall from Section 2 that by (GL n , GL r )-duality, P χ ( W ) ≃ E ⊕ dim F χ χ as G n -modules.Therefore we can assume that [ V η : E χ ] K n >
0. We will consider two separate cases.
Case I: F = R , H . Proposition 2.12 implies that [ E χ : E η ⊗ E ξ ] G n > ξ ∈ P r , andRemark 2.10 implies that η ⊆ χ , hence λ ⊆ ν . Case II: F = C . Let e χ ∈ P n be obtained from χ by adding n − r zeros on the right of the partsof χ . Our assumption entails that the restriction of the GL n ( C ) × GL n ( C )-module M e χ ⊗ M ∗ e χ to thediagonal subgroup GL n ( C ) contains the GL n ( C )-module M e η , where e η := ( m − ν r , . . . , m − ν , , . . . , , − m + ν , . . . , − m + ν r ) ∈ P n . Remark 2.10 implies that − m + ν i ≥ − m + λ i for every 1 ≤ i ≤ r , hence λ ⊆ ν . (cid:3) UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 13
The proof of our next result, Proposition 5.4, is based on facts from the theory of symmetricfunctions, which we now review quickly (for a comprehensive reference, see [18]). Let P := lim −→ k P k be the set of all partitions, where the maps P k → P k +1 are given by ( λ , . . . , λ k ) ( λ , . . . , λ k , ←− C [ x , . . . , x k ] S k denote the ring of symmetric functions, where the maps C [ x , . . . , x k +1 ] → C [ x , . . . , x k ] are givenby f f ( x , . . . , x k , h h λ , m µ i Λ := δ λ,µ ,where h λ and m µ are the complete and monomial symmetric functions associated to λ, µ ∈ P . TheSchur functions s λ , λ ∈ P , form an orthonormal basis for Λ. For every two λ, µ ∈ P such that µ ⊆ λ , the skew Schur function s λ \ µ ∈ Λ satisfies the relation h s λ \ µ , s ν i Λ = h s λ , s µ s ν i Λ for every ν ∈ P . It is well known [29, Ex. 7.56(a)] that for any skew diagram λ \ µ we have(20) s λ \ µ = s ( λ \ µ ) ◦ , where ( λ \ µ ) ◦ denotes the skew diagram obtained by a 180 degree rotation of λ \ µ . Proposition 5.4.
Let m ≥ be an integer, let m := ( m, . . . , m ) ∈ P r be the partition correspondingto the r × m rectangular Young diagram, and set P ( m ) := { λ ∈ P r : λ ⊆ m } . Let P m ( W ) be asin Definition 5.2. Then as K n -modules, P m ( W ) ≃ M µ ∈P ( m ) V µ . Proof.
Recall that K := K n is the complexification of K := K n , and that M is the complexificationof M . The map P m ( W ) → C ∞ ( X ) G r ≃ C ∞ ( Y ), f Ψ − m f , is a K -equivariant embedding. Itfollows that P m ( W ) is a direct sum of irreducible M -spherical K -modules. Next we determine themultiplicity of every K -module V λ in P m ( W ). As in Section 2, (GL n , GL r )-duality entails that P m ( W ) ≃ E m as G n -modules. We consider two separate cases. Case I: F = R , H . For every η := ( η , η , . . . , η r ) ∈ P r , we define η • ∈ P as in Section 2. ByProposition 2.12,(21) [ V λ : E m ] K n = X ξ ∈P r [ E m : E λ ⊗ E ξ ] G n . If [ E m : E λ ⊗ E ξ ] G n = 0 for some ξ ∈ P r , then [ E ∗ m : E ∗ λ ⊗ E ∗ ξ ] G n = 0 and hence λ, ξ ⊆ m (seeRemark 2.10). Therefore[ E m : E λ ⊗ E ξ ] G n = h s m • , s λ • s ξ • i Λ = h s m • \ ξ • , s λ • i Λ = h s ( m • \ ξ • ) ◦ , s λ • i Λ = δ ( m \ ξ ) ◦ ,λ . It follows that the value of (21) is 0 or 1, with the latter occurring exactly when λ ⊆ m . Case II: F = C . Let α, β ∈ P n be defined by α := ( m, . . . , m | {z } r times , , . . . ,
0) and β := ( m, . . . , m | {z } n − r times , , . . . , . The statement of the proposition is equivalent (after twisting E m by det( · ) m ⊗
1) to showing that therestriction of the GL n ( C ) × GL n ( C )-module M α ⊗ M β to the diagonal GL n ( C ) is a multiplicity-free direct sum of GL n ( C )-modules M η for η of the form(22) ( m + ξ , . . . , m + ξ r , m, . . . , m | {z } n − r times , m − ξ r , . . . , m − ξ ) , where ξ := ( ξ , . . . , ξ r ) varies through partitions satisfying ξ ≤ m . Assume that M η ⊆ M α ⊗ M β .From Remark 2.10 it follows that β ⊆ η , and in addition, every column of the skew Young diagramcorresponding to η \ β has height at most r . Since r ≤ n − r , it follows that η i ≤ m for every i > n − r .Consequently, the skew Young diagram corresponding to η \ β is a disjoint union of two (non-skew)diagrams corresponding to the partitions η + := ( η − m, . . . , η r − m ) and η − := ( η n − r +1 , . . . , η n ) . From [18, Sec. I.5.7] it follows that s η = s η + s η − , hence[ M η : M α ⊗ M β ] GL n ( C ) = h s η , s α s β i Λ = h s η \ β , s α i Λ = h s η + s η − , s α i Λ = h s η + , s α \ η − i Λ = h s η + , s ( α \ η − ) ◦ i Λ = δ η + , ( α \ η − ) ◦ . It is now straightforward to verify that [ M η : M α ⊗ M β ] GL n ( C ) ≤
1, with equality occuring if andonly if η is of the form given in (22). (cid:3) Remark 5.5.
Recall that ̺ := ( ̺ , . . . , ̺ r ) is the vector defined in (3). Let ̺ ′ ∈ C r denote thevector ̺ τ,α defined in (2), for τ := d and α := s − ̺ . Then the vectors ̺ and ̺ ′ are related by therelation ̺ r − i +1 + ̺ ′ i = s for 1 ≤ i ≤ r. Proposition 5.6.
Let λ ∈ P r , and let d λ ( x, s ) be as in Lemma 5.1. Let ̺ τ,α be as in (2) for τ := d and α := s − ̺ . For every ν ∈ P r , if λ * ν then (23) d λ (cid:0) ν + ̺ τ,α , s (cid:1) = 0 for every s ∈ C . Proof.
Fix ν ∈ P r . Since d λ ( x, s ) is a polynomial in x = ( x , . . . , x r ) and s , it is enough to prove(23) for infinitely many values of s . Since d λ is even-symmetric, vanishing of d λ ( x, s ) at a point( x , . . . , x r ) is equivalent to its vanishing at the point ( − x r , . . . , − x ). Therefore it suffices to showthat d λ ( y, s ) = 0 for y := ( y , . . . , y r ) given by y i := − ν r +1 − i − d i − − s + ̺ for every 1 ≤ i ≤ r. Next set s := − m for some integer m ≥ λ . By Remark 5.5 and Lemma 5.1, it suffices to show that(24) c λ, − m ( η ) = 0 for η := ( m − ν r , . . . , m − ν ) . Recall that m := ( m, . . . , m ) ∈ P r is the partition corresponding to an r × m Young diagram. Set(25) R m ( W ) := (cid:8) Ψ − m f : f ∈ P m ( W ) (cid:9) , where Ψ is defined in Section 3. The map j m : R m ( W ) → P m ( W ) , f Ψ m f is an isomorphism of K n -modules. Therefore Proposition 5.4 implies that V η occurs as a K n -submodule of R m ( W ). By restriction to X , and then factoring G r -invariant functions on X through UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 15 Y ≃ X/G r , we obtain an embedding R m ( W ) ⊂ C ∞ ( Y ). From K n -invariance of D λ, − m it followsthat D λ, − m R m ⊆ R m . Furthermore, the diagram R m ( W ) D λ, − m / / j m (cid:15) (cid:15) R m ( W ) j m (cid:15) (cid:15) P m ( W ) D λ / / P m ( W )commutes. From the latter commutative diagram it follows that, in order to prove (24), it sufficesto show that ( D λ ◦ j m ) V η = { } . Note that D λ ∈ ι ( F λ ⊗ F ∗ λ ), where ι is the map defined in (13).The elements of ι ( F ∗ λ ) act on P ( W ) as K n -invariant constant coefficient differential operators, andby considering the G r -action it follows that they map P m ( W ) into P χ ( W ), where χ = ( m − λ r , . . . , m − λ ) . Therefore the map D λ : P m ( W ) → P m ( W ) factors through a K n -equivariant map(26) P m ( W ) → P χ ( W ) . Suppose that ( D λ ◦ j m ) V η = { } . Since the map (26) is K n -equivariant, [ V η : P χ ( W )] K n >
0. FromProposition 5.3 it follows that λ ⊆ ν , which is a contradiction. (cid:3) Proof of Theorem 1.2
In this section we complete the proof of Theorem 1.2. We start by proving the following lemma.
Lemma 6.1.
Let λ ∈ P r . Then there exists a constant γ ′ λ ∈ C such that for every s ∈ C , and every µ ∈ P r , (27) c λ,s ( µ ) = γ ′ λ P λ (cid:18) µ + ̺ ; d , s − ̺ (cid:19) . Proof.
First fix s > ̺ . From Proposition 5.6 and Lemma 5.1 we obtain c λ,s (cid:0) ν + (cid:0) d ( r −
1) + s − ̺ (cid:1) (cid:1) = 0 if | ν | ≤ | λ | and ν = λ. From the vanishing part of the statement of Proposition 4.3 it follows that the polynomial q λ ( x ) := P λ (cid:0) x + ̺ ; d , s − ̺ (cid:1) also vanishes for all x := ν + (cid:0) d ( r −
1) + s − ̺ (cid:1) , where ν satisfies | ν | ≤ | λ | and ν = λ . The uniqueness part of the statement of Proposition 4.3 now implies that there exists ascalar γ ′ λ ∈ C , possibly depending on s , such that (27) holds for every µ . Since both sides of (27)are polynomials in s and µ (see Lemma 5.1), and (27) holds for a Zariski dense subset of values( µ, s ) ∈ C r × C , it follows that (27) indeed holds for every s ∈ C , and γ ′ λ is a rational function of s ∈ C .Next we show that γ ′ λ does not depend on s . Since the value of c λ,s ( µ ) is a polynomial in s and µ , and P λ ( µ + ̺ ; d , s − ̺ ) is 2 λ -monic in µ , it follows that γ ′ λ is a polynomial in s . However,from the combinatorial formula for P λ that is given in (16), it follows that the degree of s in P λ (cid:0) µ + ̺ ; d , s − ̺ (cid:1) is exactly 2 | λ | , whereas the degree of s in c λ,s ( µ ) is at most 2 | λ | . By comparingthe degrees of s on both sides of (27), it follows that γ ′ λ is a constant independent of s . (cid:3) To complete the proof of Theorem 1.2, we need to prove that γ ′ λ = γ λ , where γ λ is defined in(18). The rest of this section is devoted to the proof of the latter claim.Set t := dim R ( A ) and let v , . . . , v t be an orthonormal basis for A with respect to the pairing( · , · ) A . Set ϕ i := ( · , v i ) A ∈ A ∗ for every 1 ≤ i ≤ t , so that ϕ i ( v j ) = δ i,j . For every 1 ≤ i ≤ t , let ∂ v i ∈ D ( A ) denote the directional derivative corresponding to v i , so that ∂ v i ( ϕ j ) = ϕ j ( v i ) = δ i,j . Forevery 1 ≤ i ≤ t , set q i := ψ ∗ ( ϕ i ) where ψ ∗ is as in (6), and let ∂ q i ∈ D ( W ) denote the second-orderdifferential operator corresponding to q i under the isomorphism P ( W ) ≃ D ( W ) defined in (12).Fix an integer m ≥ D ( m ) := X | λ | = m e D λ , where e D λ is defined in (15). Then D ( m ) acts on every K -module V µ ⊆ C ∞ ( Y ) ≃ C ∞ ( X ) G r by thescalar c ( m ) ( µ ) := X | λ | = m c λ, ( µ ) . From the definition of the operators e D λ it follows that if { e v j } is a basis for P m ( A ) and { e ∂ j } is thecorresponding dual basis for D m ( A ) ≃ P m ( A ) ∗ , then D ( m ) = P j ι ( e v j ) ι ( e ∂ j ) where ι is as in (13).In particular, if we choose the basis (cid:8) ( ∂ v ) m · · · ( ∂ v t ) m t : m + · · · + m t = m (cid:9) for D m ( A ), then the corresponding dual basis for P m ( A ) will be (cid:26) m ! · · · m t ! ϕ m · · · ϕ m t t : m + · · · + m t = m (cid:27) , and thus D ( m ) = 1 m ! X m + ··· + m t = m (cid:18) mm , . . . , m t (cid:19) q m · · · q m t t ( ∂ q ) m · · · ( ∂ q t ) m t = 1 m ! ( q ∂ q + · · · + q t ∂ q t ) m + D ′ = 1 m ! (cid:0) D (1) (cid:1) m + D ′ , where D ′ is a differential operator with order strictly less than the order of D .Let g C be the Lie algebra of G n , and let Ω k and Ω g denote the Casimir operators of k C and g C (see the Appendix for more information). Also, Let E be the degree (or Euler) operator on W .The operator D (1) is a polynomial-coefficient differential operator on W C . The next propositionexpresses D (1) in terms of Ω k , Ω g , and E . Proposition 6.2.
Let D (1) be defined as in (28) . Then (29) D (1) = − n − k + Ω g − E if F = R , − k + 2Ω g + (2 n − r ) E if F = C , − n + 1)Ω k + 2Ω g + 2(2 n − r + 1) E if F = H . Proof.
The proof is by a tedious calculation and is deferred to the Appendix. (cid:3)
Lemma 6.3.
The scalar c (1) ( µ ) is a quadratic polynomial in µ , . . . , µ r with top-degree homogeneouspart equal to − (cid:0) µ + · · · + µ r (cid:1) .Proof. We use the expressions for D (1) given in Proposition 6.2. As is well known (e.g., see [4, Lem.3.3.8]), the action of Ω k on V µ is by a scalar which is a quadratic polynomial in µ , . . . , µ n , whose UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 17 top homogeneous term is given by(30) n − ( µ + · · · + µ r ) if F = R , µ + · · · + µ r ) if F = C , n +2 ( µ + · · · + µ r ) if F = H . Fix m ∈ Z such that m ≥ µ , and let R m ( W ) be as in (25). From Proposition 5.4 it followsthat [ V µ : R m ( W )] K n >
0. Since elements of R m ( W ) are homogeneous of degree zero, the degreeoperator E vanishes on V µ . Furthermore, we have V µ ⊆ C ∞ ( Y ) K n -finite , and Y ≃ G n /P r,n where P r,n is the ( r, n − r ) parabolic subgroup of G n . Thus, C ∞ ( G n /P r,n ) K n -finite is the space of K n -finitevectors of a degenerate principal series representation of G n induced from P r,n , and the operator Ω g acts on the latter space by a scalar that is independent of µ (see [10, Prop. 8.22]). Consequently, thetop-degree homogeneous part of c (1) ( µ ) is determined by the action of Ω k . The lemma now followsfrom (29) and (30). (cid:3) Lemma 6.3 implies that the action of D ( m ) on V µ is by a polynomial in µ , . . . , µ r of degree 2 m ,whose top-degree homogeneous part is ( − m m ! ( µ + · · · + µ r ) m . On the other hand, from [14, Eq.(7.3)] it follows that for every λ such that | λ | = m , the top-degree homogeneous part of c λ, ( µ ) is upto a scalar equal to P λ ( µ , d ), where P λ ( x, τ ) is the λ -monic Jack polynomial and µ := ( µ , . . . , µ r ).Let J λ denote the normalization of the Jack polynomial introduced in [28, Thm 1.1]. The scalarrelating J λ and P λ is given in [18, Chap. VI, Eq. (10.22)] (see also [28, Thm 5.6]). From [28, Prop.2.3], and the relation between P λ and J λ , it follows that(31) ( µ + · · · + µ r ) m = X | λ | = m (cid:18) d (cid:19) m m ! Q ♭ ∈ λ (cid:0) d ( a λ ( ♭ ) + 1) + l λ ( ♭ ) (cid:1) P λ (cid:18) µ , d (cid:19) . Since the polynomials P λ ( µ , d ) are linearly independent, by considering the top-degree homogeneousparts of both sides of (31) it follows that for every λ ∈ P r such that | λ | = m , c λ, ( µ ) = ( − d ) m Q ♭ ∈ λ (cid:0) d ( a λ ( ♭ ) + 1) + l λ ( ♭ ) (cid:1) P λ (cid:18) µ + ̺ ; d , − ̺ (cid:19) . Lemma 6.1 completes the proof of Theorem 1.2.
Appendix: Proof of Proposition 6.2
In this Appendix, we exhibit the details of the calculations that yield the formulas (29) for theoperator D (1) .Let κ : k C × k C → C denote the invariant bilinear form which is equal to the Killing form of k C when F = R or H , and is given by κ ( x, y ) := tr( xy ) when F = C . The Casimir operator of k C isΩ k := P dim k C i =1 x i x i , where { x i : 1 ≤ i ≤ dim k C } is a basis for k C , and { x i : 1 ≤ i ≤ dim k C } is thecorresponding dual basis with respect to κ ( · , · ). We define the Casimir operator Ω g of g C similarly.Explicit formulas for Ω g are given in (33), (36), and (38). In the following, E i,j will always denotea matrix with a 1 in the ( i, j ) position and 0’s elsewhere (the number of rows and columns of E i,j will be clear from the context). Case I: F = R . Recall that in this case A = Sym r × r ( R ). The orthonormal basis of A with respectto ( · , · ) A is E i,i for 1 ≤ i ≤ r and 1 √ E i,j + E j,i ) for 1 ≤ i < j ≤ r. We fix generators x i,j ∈ P ( A ), where 1 ≤ i ≤ j ≤ r , and y i,j ∈ P ( W ), where 1 ≤ i ≤ n and1 ≤ j ≤ r , such that x i,j ([ t a,b ]) := t i,j for every matrix [ t a,b ] ∈ A , and y i,j ([ t a,b ]) := t i,j for everymatrix [ t a,b ] ∈ W . The isomorphism ι A : A → A ∗ ≃ P ( A ) that is defined in (11) is given by E i,i x i,i for 1 ≤ i ≤ r and 1 √ E i,j + E j,i )
7→ √ x i,j for 1 ≤ i < j ≤ r. The map ψ ∗ : P ( A ) → P ( W ) of (6) is given by x a,a n X i =1 y i,a for 1 ≤ a ≤ r and √ x a,b
7→ √ r X i =1 y i,a y i,b for 1 ≤ a < b ≤ r. Finally, the isomorphism ι W : W ≃ W ∗ ≃ D ( W ) of (11) is given by y i,a ∂ i,a := ∂∂y i,a . From allof the above, it follows that(32) D (1) = r X a =1 n X i =1 y i,a ! n X j =1 ∂ j,a + X ≤ a = b ≤ r n X i =1 y i,a y i,b ! n X j =1 ∂ j,a ∂ j,b . The embedding k C ֒ → gl n ( C ) gives the realization of k C as k C = (cid:8) x ∈ Mat n × n ( C ) : x + x T = 0 (cid:9) , where x T is the transpose of x . Recall that by the definition of Ω g ,(33) Ω g = X ≤ i,j ≤ n E i,j E j,i . The Killing form of k C is κ ( x, y ) := ( n − xy ), and therefore(34) Ω k = − n − X ≤ i The map ψ ∗ : P ( A ) → P ( W ) of (6) is given by ψ ∗ ( ι A ( S )) := P na =1 x a,i + y a,i if S = E i,i where 1 ≤ i ≤ r, √ P na =1 ( x a,i x a,j + y a,i y a,j ) if S = √ ( E i,j + E j,i ) where 1 ≤ i < j ≤ r, √ P na =1 ( − x a,j y a,i + x a,i y a,j ) if S = √− √ ( E i,j − E j,i ) where 1 ≤ i < j ≤ r. For the realization of the derived action of G n ≃ GL n ( C ) × GL n ( C ) on P ( W ) it will be moreconvenient to work with the coordinates z i,j and ξ i,j on W C ≃ Mat n × r ( C ) ⊕ Mat n × r ( C ), where1 ≤ i ≤ n and 1 ≤ j ≤ r , given by z i,j := x i,j − √− y i,j and ξ i,j := x i,j + √− y i,j . From these formulas it follows that ∂∂x i,j = ∂∂z i,j + ∂∂ξ i,j and ∂∂y i,j = √− (cid:16) ∂∂ξ i,j − ∂∂z i,j (cid:17) .Set ∂ ξ i,j := ∂∂ξ i,j and ∂ z i,j := ∂∂z i,j . By a direct calculation, we obtain D (1) = 4 r X i =1 n X a =1 z a,i ξ a,i ! n X a =1 ∂ z a,i ∂ ξ a,i ! + 2 X ≤ i 4, set ∂ Φ e ( a, b ) := ( ι W ) − (Φ e ( a, b )), where ι W : D ( W ) → P ( W ) is the isomorphismgiven in (12). In fact ∂ Φ e ( a, b ) is the constant coefficient differential operator obtained from Φ e ( a, b )by substitution of each variable ξ i,j by the corresponding partial derivative ∂ ξ i,j . Then D (1) = 4 X ≤ a,b ≤ r (Φ ( a, b ) − Φ ( a, b )) ( ∂ Φ ( a, b ) − ∂ Φ ( a, b ))+ 2 X ≤ a,b ≤ r (Φ ( a, b ) − Φ ( b, a )) ( ∂ Φ ( a, b ) − ∂ Φ ( b, a ))+ 2 X ≤ a,b ≤ r (Φ ( a, b ) − Φ ( b, a )) ( ∂ Φ ( a, b ) − ∂ Φ ( b, a )) . The embedding of k C into the Lie algebra of G n gives the realization of k C as k C := (cid:8) x ∈ Mat n × n ( C ) : x T J n = − J n x (cid:9) , where J n is as in (5), and x T denotes the transpose of the matrix x . We will denote the matricesin the standard basis of gl n ( C ) by E i,j ’s. The Killing form of k C is κ ( x, y ) := (4 n + 2)tr( xy ), and UADRATIC CAPELLI OPERATORS AND OKOUNKOV POLYNOMIALS 21 consequently,Ω k = 14(2 n + 1) X ≤ p,q ≤ n ( E p,q − E q + n,p + n ) ( E q,p − E p + n,q + n )+ 14(2 n + 1) X ≤ p Schur Q -functions and the Capelli eigenvalue problem for the Lie superal-gebra q ( n ). 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