We call a Maass lift a prescription to associate an irreducible admissible resp. uni-tary representation of G = PGSp(4) to an irreducible admissible resp. unitary representation of the Jacobi group G J = SL(2) n H. Such a prescription is constructed in the general local case for the principal series representations J m;; of G J and moreover in the real case for the discrete series representatio...

The irreducible unitary representations of GL(n, ~) have been classified by D. Vogan in [V2]. In his work, four families of representations play a special role. They are (i) the one dimensional unitary representations and (ii) their complementary series; (iii) Speh's representations and (iv) their complementary series. These representations serve as "building blocks" in the sense that every irr...

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.

Title of dissertation: MATROIDS AND GEOMETRIC INVARIANT THEORY OF TORUS ACTIONS ON FLAG SPACES Benjamin Howard, Doctor of Philosophy, 2006 Dissertation directed by: Professor John Millson Department of Mathematics Let λ and μ be weights of G = SL(n,C) such that λ is dominant. Let Vλ be the irreducible representation of G with highest weight λ, and let Vλ[μ] denote the μ-th weight space within V...

Let G be a connected unipotent group over a finite field Fq. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(Fq) and give an explicit description of L-packets in terms of the so-called “admissible pairs” for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension...

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. The result is applied to prove a conjectural character formula put forward by van der Jeugt et al in the late 80s. We simplify this character formula to cast it into the Kac-Weyl form, and derive from it a closed formula for the dimension...

Let G be a …nite group and be a faithful irreducible character for G. Earlier papers by the …rst author describe techniques for computing a matrix representation for G which a¤ords whenever the degree (1) is less than 32. In the present paper we introduce a new, fast method which can be applied in the important case where G is perfect and the socle soc(G=Z(G)) of G over its centre is abelian. I...

1.1. Shan has proved that the categories Oc(Wn) for rational Cherednik algebras of type Wn = W (G(`, 1, n)) = Snn(μ`) with n varying, together with decompositions of the parabolic induction and restriction functors of Bezrukavnikov-Etingof, provide a categorification of an integrable s̃le Fock space representation F(m), [18]. The parameters m ∈ Z` and e ∈ N ∪ {∞} arise from the choice of paramet...

In this paper we study the branching problems for Hecke algebra H(Dn) of type Dn. We explicitly describe the decompositions of the socle of the restriction of each irreducible H(Dn)-representation to H(Dn−1) into irreducible modules by using the corresponding results for type B Hecke algebras. In particular, we show that any such restrictions are always multiplicity free.