Hermitian Symmetric Pairs Research Articles

Dirac cohomology for the BGG category [formula omitted

We study Dirac cohomology HDg,h(M) for modules belonging to category O of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of M⊗S, with S being a spin module of h⊥. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for HDg,h(M) while we show that in the case of a Hermitian symmetric pair (g,k) and an irreducible unitary module M∈O, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in M. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for M∈O.

Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.

Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

Abstract Let $(G,K)$ be an irreducible Hermitian symmetric pair of non-compact type with $G={SU}(p,q)$, and let $\lambda$ be an integral weight such that the simple highest weight module $L(\lambda)$ is a Harish-Chandra $({\mathfrak{g}},K)$-module. We give a combinatorial algorithm for the Gelfand–Kirillov (GK) dimension of $L(\lambda)$. This enables us to prove that the GK dimension of $L(\lambda)$ decreases as the integer $\langle{\lambda+\rho},{\beta}^{\vee} \rangle$ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of $L(\lambda)$.

CLOSED ORBITS ON PARTIAL FLAG VARIETIES AND DOUBLE FLAG VARIETY OF FINITE TYPE

Let $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^{\theta})_{0} $ the identity component of the fixed points of an involutive automorphism $ \theta $ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield to conclude that the $K$-orbit through the origin $(e P_{1}, e P_{2})$ of $G/P_{1} \times G/P_{2}$ is closed and it generates an open dense $G$-orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed $K$-orbits on $G/P_{1} \times G/P_{2}$.

Super Duality and Homology of Unitarizable Modules of Lie Algebras

The \mathfrak u -homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types \mathfrak gl (n) , \mathfrak sp (2n) and \mathfrak so (2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU(p,q),SU(p)\times SU(q)) , (Sp(2n),U(n)) and (SO^\ast(2n),U(n)) .

On the Ext algebras of parabolic Verma modules and [formula omitted]-structures

We study the Ext-algebra of the direct sum of all parabolic Verma modules in the principal block of the Bernstein–Gelfand–Gelfand category O for the Hermitian symmetric pair ( g l n + m , g l n ⊕ g l m ) and present the corresponding quiver with relations for the cases n = 1 , 2 . The Kazhdan–Lusztig combinatorics is used to deduce a general vanishing result for the higher multiplications in the A ∞ -structure of a minimal model. An example of higher multiplications with non-vanishing m 3 is included.

On the principal symbols of $K_C$- invariant differential operators on Hermitian symmetric spaces

Let $(G,K)$ be one of the following Hermitian symmetric pair: $SU(p,q),S(U(p) \: \times \: U(q)))$, $(Sp(n,$\mathbf{R}$ ,U(n))$, or $(SO^*(2n),U(n))$. Let $G_C$ and $K_C$ be the complexifications of $G$ and $K$, respectively, $Q$ the maximal parabolic subgroup of $G_C$ whose Levi part is $K_C$, and $V$ the holomorphic tangent space at the origin of $G/K$. It is known that the ring of $K_C$ -invariant differential operators on $V$ has a generating system $\{\mathit{\Gamma_k}\}$ given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result is that determinant or Pfaffian of a deformation of the twisted moment map on the holomorphic cotangent bundle of $G_C / Q$ provides a generating function for the principal symbols of $\mathit{\Gamma_k}$'s.

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u -homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u -homology computation are worked out.

Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan–Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan–Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan–Lusztig polynomials of certain Weyl groups.

Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients

Abstract. Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.

Parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs

We study the parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. In particular, we show that these polynomials are always either zero or a monic power of $q$, and that they are combinatorial invariants.

Intersection of harmonics and Capelli identities for symmetric pairs

We consider a see-saw pair consisting of a Hermitian symmetric pair ( G R , K R ) and a compact symmetric pair ( M R , H R ) , where ( G R , H R ) and ( K R , M R ) form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain K C -invariant elements in U( g C ) in terms of H C -invariant elements in U( m C ) . The corresponding H C -invariant elements are called Capelli elements. We also give a decomposition of the intersection of O 2n -harmonics and S p 2n -harmonics as a module of G L n = O 2n ∩S p 2n , and construct a basis for the G L n highest weight vectors. This intersection is in the kernel of our Capelli elements.

Del Pezzo surfaces and representation theory

The connection between del Pezzo surfaces and root systems goes back to Coxeter and Du Val, and was given modern treatment by Manin in his seminal book Cubic forms. Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded into a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. Computational proofs of this conjecture based on the structure of the Cox ring have been given recently by Popov and Derenthal. We give a new proof of Batyrev’s conjecture using an inductive process, interpreting the blowing-up of a point on a del Pezzo surface in terms of representations of Lie algebras corresponding to Hermitian symmetric pairs.

Parabolic Kazhdan–Lusztig R-polynomials for Hermitian symmetric pairs

We give explicit combinatorial product formulas for the parabolic Kazhdan–Lusztig R-polynomials of Hermitian symmetric pairs. Our results imply that all the roots of these polynomials are (either zero or) roots of unity, and complete those in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Young's lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on Hermitian symmetric pairs of type A. As an application of our results, we derive explicit combinatorial product formulas for certain sums and alternating sums of ordinary Kazhdan–Lusztig R-polynomials.

SPHERICAL FUNCTIONS, THE COMPLEX HYPERBOLIC PLANE AND THE HYPERGEOMETRIC OPERATOR

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = ( SU (3), U (2)) and ( SU (2,1), U (2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair ( SU (3), U (2)), this project was started in [4].

A Pieri rule for Hermitian symmetric pairs I

Let X be an irreducible Hermitian symmetric space of non-compact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m 1 ...,m r ) with m 1 >... > m r > 0. That is, the same parameters as the irreducible polynomial representations of GL(r). Let S[m 1 ,...,m r ] be the corresponding isotypic component. In this article we show that the product in S, S[m 1 ,...,m r ]S[k,0,0,...,0] is a direct sum of constituents following precisely the classical Pieri rule.

Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules

Let ( G, K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g= k⊕ p +⊕ p − be the usual decomposition of g as a k -module. There is a natural correspondence between K C -orbits in p + and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Y k the closure of the K C -orbit in p + that is associated to the kth Wallach representation. In this article we give explicit formulas for the numerator polynomials of the Hilbert series of the varieties Y k by using BGG resolutions of unitarizable highest weight modules. A preliminary result gives a new branching formula for a certain two-parameter family of finite dimensional representations of the even orthogonal groups. Our work is an extension of previous work by Enright and Willenbring.

Closed product formulas for extensions of generalized Verma modules

We give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of ( g , p ) (g,p) -generalized Verma modules, in the cases when ( g , p ) (g,p) corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic R R -polynomials introduced by Deodhar.

The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair

The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair

Derived Functors of Unitary Highest Weight Modules at Reduction Points

The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the ${\mathbf {K}}$-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest ${\mathbf {K}}$-type are also easily computed.