Contragredient representations and characterizing the local Langlands correspondence

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R)

Hecke algebras and involutions in Weyl groups

For any two involutions y,w in a Weyl group (y\le w), let P_{y,w} be the polynomial defined in [KL]. In this paper we define a new polynomial P^\sigma_{y,w} whose i-th coefficient is a_i-b_i where the i-th coefficient of P_{y,w} is a_i+b_i (a_i,b_i are natural numbers). These new polynomials are of interest for the theory of unitary representations of complex reductive groups. We present an algorithm for computing these polynomials.Comment: 25 page

Computing the associatied cycles of certain Harish-Chandra modules

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. In \cite{MPVZ} we proved that for any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2}\dim(G_{\mathbb{R}}/K_{\mathbb{R}})$, the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$ is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly

Functions on the model orbit in E8

First published in Representation Theory in Vol.2,1998. Published by the American Mathematical Society.We decompose the ring of regular functions on the unique coadjoint orbit for complex E8 of dimension 128, finding that every irreducible representation appears exactly once. This confirms a conjecture of McGovern. We also study the unique real form of this orbit

Strictly small representations and a reduction theorem for the unitary dual

First published in Representation Theory in Vol 5, 2001. Published by the American Mathematical Society.To any irreducible unitary representation X of a real reductive Lie group we associate in a canonical way, a Levi subgroup Gsu and a representation of this subgroup. Assuming a conjecture of the authors on the infinitesimal character of X, we show that X is cohomologically induced from a unitary representation of the subgroup Gsu. This subgroup is in some cases smaller than the subgroup Gu that the authors attached to X in earlier work. In those cases this provides a further reduction to the classification problem