Irreducible Harish-Chandra Modules Research Articles

Regular Functions on the 𝐾-nilpotent cone

Let G G be a complex reductive algebraic group with Lie algebra g \mathfrak {g} and let G R G_{\mathbb {R}} be a real form of G G with maximal compact subgroup K R K_{\mathbb {R}} . Associated to G R G_{\mathbb {R}} is a K × C × K \times \mathbb {C}^{\times } -invariant subvariety N θ \mathcal {N}_{\theta } of the (usual) nilpotent cone N ⊂ g ∗ \mathcal {N} \subset \mathfrak {g}^* . In this article, we will derive a formula for the ring of regular functions C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K × C × K \times \mathbb {C}^{\times } . Some motivation comes from Hodge theory. In [Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences, Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito’s theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G R G_{\mathbb {R}} is split, and X X is the spherical principal series representation of infinitesimal character 0 0 , then conjecturally g r ( X ) ≃ C [ N θ ] gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K × C × K \times \mathbb {C}^{\times } . So a formula for C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations.

Dirac series for complex classical Lie groups: A multiplicity-one theorem

This paper computes the Dirac cohomology HD(π) of irreducible unitary Harish-Chandra modules π of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free K-type in π contributing to HD(π). This confirms conjectures formulated by the first named author and Pandžić in 2011.

Sign signatures and characters of Weyl Groups

Let GR be the set of real points of a complex linear reductive group and Gˆλ, its classes of irreducible admissible representations with infinitesimal integral regular character λ. In this case, each cell of representations is associated to a special nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to bypass the need for the character table of the Weyl group associated with Gˆλ in the Springer correspondence, by describing a new and less computationally intensive parametrization of irreducible representations of simple complex Weyl groups.

Equivariant deformation quantization and coadjoint orbit method

Our goal here is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive Lie groups. We first prove some general results on the existence of equivariant deformation quantizations of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to the orbit method and construct nontrivial irreducible Harish-Chandra modules for certain nilpotent coadjoint orbits. Our examples include new geometric construction of representations associated to a large class of nilpotent orbits of real exceptional Lie groups.

Classification of irreducible Harish-Chandra modules over gap-p Virasoro algebras

We prove that any irreducible Harish-Chandra module for a class of Lie algebras, which we call gap-p Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series. These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra Vir, but graded by pZ (unlike Vir by Z).

Reducible characteristic cycles of Harish-Chandra modules for and the Kashiwara–Saito singularity

We give examples of reducible characteristic cycles for irreducible Harish-Chandra modules for U(p, q) by analyzing a four-dimensional singular subvariety of We relate this singularity to the Kashiwara–Saito singularity arising for Schubert varieties for with reducible characteristic cycles, as well as recent related examples of Williamson. In particular, this gives another explanation of the simple highest weight modules for with reducible associated varieties that Williamson discovered.

Unitary Representations with Dirac Cohomology: Finiteness in the Real Case

Abstract Let $G$ be a complex connected simple algebraic group with a fixed real form $\sigma $. Let $G({\mathbb{R}})=G^\sigma $ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of $G({\mathbb{R}})$ (up to equivalence) having nonvanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain $G({\mathbb{R}})$, with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.

The explicit theta correspondence for reductive dual pairs (Sp(p,q),O⁎(4))

For every irreducible Harish–Chandra module of O⁎(4), we determine its theta lifts to Sp(p,q) in terms of the Langlands parameters, for all non-negative integers p and q. Our strategy is to determine the desired theta lifts by their infinitesimal characters and lowest K-types.

Codimension one connectedness of the graph of associated varieties

Let $ \pi $ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ \mathcal{AV}(\pi) $. If $ \mathcal{AV}(\pi) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ \pi $, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connected for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ \Gamma_K(\mathcal{O}^G_\lambda) $ thus defined, there is an irreducible Harish-Chandra module $ \pi $ whose associated graph is exactly equal to the connected component.

Harish-Chandra modules over generalized Heisenberg-Virasoro algebras

In this paper, we give a complete classification of irreducible Harish-Chandra modules for any generalized Heisenberg-Virasoro algebra. In particular, we present a simpler and more conceptual proof of the classification of irreducible Harish-Chandra modules over the classical Heisenberg-Virasoro algebra, which was first obtained by Rencai Lu and Kaiming Zhao in [LZ1]. Our methods are based on the ideas of polynomial modules from [B1, BB].

Representations of the Extended W-Algebra

In this article, classification of all irreducible Harish–Chandra modules over (the universal central extension of W(α, β)) for a i = 0, b i ≠ 1, i = 1, 2 is given. In addition, classification of irreducible weight modules with a finite dimensional weight space over for a i = 0, − 1 ≤ b i ≠ 1 if b i ∈ ℤ, i = 1, 2 is also obtained. Finally, we study the irreducibility of Verma modules over a subalgebra of .

Irreducible Harish-Chandra modules over extended Witt algebras

Let d be a positive integer, \(A={\mathbb{C}} [t_{1}^{\pm1},\ldots ,t_{d}^{\pm1}]\) be the Laurent polynomial algebra, and \(W=\operatorname{Der} (A)\) be the derivation Lie algebra of A. Then we have the semidirect product Lie algebra W⋉A which we call the extended Witt algebra of rank d. In this paper, we classify all irreducible Harish-Chandra modules over W⋉A with nontrivial action of A.

Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

AbstractLet G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

HARISH-CHANDRA MODULES OVER THE ℚ HEISENBERG–VIRASORO ALGEBRA

AbstractIn this paper, it is proved that all irreducible Harish-Chandra modules over the ℚ Heisenberg–Virasoro algebra are of the intermediate series (all weight spaces are at most one-dimensional).

Simple Harish-Chandra modules, intermediate series modules, and Verma modules over the loop-Virasoro algebra

This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also establish an irreducibility criterion for Verma modules over the loop Virasoro algebra and determine necessary and sufficient conditions for the simple quotient of a Verma module to be a Harish-Chandra module.

Dirac Cohomology of some Harish-Chandra Modules

In this paper we determine the Dirac cohomology of certain irreducible Harish-Chandra modules of a semisimple connected Lie group G with finite center: irreducible finite-dimensional modules and unitary A $\mathfrak{q}$ (λ) modules. We also comment on the relationship to ( $\mathfrak{g}$ , K)-cohomology.

Signatures of Invariant Hermitian Forms on Irreducible Highest-Weight Modules

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com- position factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig Conjecture which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polyno- mials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant for- mula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest weight module. Such a formula has applications to unitarity testing. 1.1. The Unitary Dual Problem. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif- ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra. Fourier analysis is just one incarnation of this programme. An unresolved component in Gelfand's programme is the classification of the irre- ducible unitary representations of a group, known as the unitary dual problem. In the case of a real reductive Lie group, the problem is equivalent to identifying all irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form. As Harish-Chandra modules may be constructed via an algebraic method introduced by Zuckerman in 1978 known as cohomological induction, it is of interest to study signatures of invariant Hermitian forms on cohomologically induced modules and to understand how positivity can fail. Cohomological induction is a two-step process in which we compose an induction functor with a Zuckerman functor i . The intermediate module in cohomological induction is a generalized Verma module which admits an invariant Hermitian form if the module to which induction was applied admits an invariant Hermitian form. Formulas for the signatures of invariant Hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo- logically induced modules (eg. (Wal84)). This motivates the study of invariant Hermitian forms on Verma modules (see (Wal84), (Yee05)) and irreducible highest weight modules.

K-finite matrix elements of irreducible Harish-Chandra modules are hypergeometric functions

We show that all the K-finite matrix elements of irreducible Harish-Chandra modules can can be expressed in spherical functions using finite number of operations

On the centralizer of K in [formula omitted

Let g = k + p be a complexified Cartan decomposition of a complex semisimple Lie algebra g and let K be the subgroup of the adjoint group of g corresponding to k . If H is an irreducible Harish-Chandra module of U ( g ) , then H is completely determined by the finite-dimensional action of the centralizer U ( g ) K on any one fixed primary k component in H. This original approach of Harish-Chandra to a determination of all H has largely been abandoned because one knows very little about generators of U ( g ) K . Generators of U ( g ) K may be given by generators of the symmetric algebra analogue S ( g ) K . Let S m ( g ) K , m ∈ Z + , be the subalgebra of S ( g ) K defined by K-invariant polynomials of degree at most m. For convenience write A = S ( g ) K and A m for the subalgebra of A generated by S m ( g ) K . Let Q and Q m be the respective quotient fields of A and A m . We prove that if n = dim g one has Q = Q 2 n . We also determine the variety, Nil K , of unstable points with respect to the action K on g and show that Nil K is already defined by A 2 n . As pointed out to us by Hanspeter Kraft, this fact together with a result of Harm Derksen (see [H. Derksen, Polynomial bounds for rings of invariants, Proc. Amer. Math. Soc. 129 (4) (2001) 955–963]) implies, indeed, that A = A r where r = ( 2 n 2 ) dim p .

Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting

We establish an algorithm to compute characters of irreducible Harish-Chandra modules for a large class of nonalgebraic Lie groups. (Roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in Harish-Chandra's class.) We then apply this theory to study a particular group (the universal cover of the real general linear group), and discover a symmetry of the character computations encoded in a character multiplicity duality. Using this duality theory, we reinterpret a kind of representation-theoretic Shimura correspondence for the general linear group geometrically, and find that it is dual to an analogous lifting for indefinite unitary groups. It seems likely that this example is illustrative of a general framework for studying similar correspondences.