A generating function for the trace of the Iwahori-Hecke algebra

The Iwahori-Hecke algebra has a ``natural'' trace $\tau$. This trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important commutative sub-algebra ${\bf C}[\theta_x]$, that was described and studied by Bernstein, Zelevinski and Lusztig. In this note we compute the generating function for the value of $\tau$ on the basis $\theta_x$

Periodic integrable systems with delta-potentials

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta function interactions. The underlying symmetry structures are shown to be governed by the associated graded of Cherednik's (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkin's generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.Comment: 36 pages. The analysis of the propagation operator in Sections 5 and 6 is corrected and simplified. To appear in Comm. Math. Phy

The infinitesimal characters of discrete series for real spherical spaces

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of $H$.Comment: To appear in GAF

Algebraic and analytic Dirac induction for graded affine Hecke algebras

We define the algebraic Dirac induction map \Ind_D for graded affine Hecke algebras. The map \Ind_D is a Hecke algebra analog of the explicit realization of the Baum-Connes assembly map in the $K$-theory of the reduced $C^*$-algebra of a real reductive group using Dirac operators. The definition of \Ind_D is uniform over the parameter space of the graded affine Hecke algebra. We show that the map \Ind_D defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded Hecke algebra analogue of the construction of discrete series representations for semisimple Lie groups due to Parthasarathy and Atiyah-Schmid.Comment: 37 pages, revised introduction, updated references, minor correction

SINGULARITIES OF HOLOMORPHICALLY EXTENDED SPHERICAL FUNCTIONS

1. Motivation and background\ud This is a preliminary account of joint work in progress which serves as an (very) extended abstract for a presentation of one of us ( $E.M$ .O.) in the Awajishima conference on Representation Theory, November 16-19, 2004, Japan.\ud To motivate the questions which we will address we will describe in some detail the beautiful ideas that were initiated by Sarnak [18] and by Bernstein and Reznikov [2], and then further explored by Kr\"otz and Stanton [9].\ud Inspired by Sarnak [18], Bernstein and Reznikov [2] proposed a new method for estimating the coefficients in the expansion of the square of a Maass form on a compact locally symmetric space $Z=\Gamma\backslash X$ (where $X=G/K$ denotes a noncompact Riemannian symmetric space, and\ud $\Gamma\subset G$ is a $co$-compact discrete subgroup of $G$ ) with respect to an orthonormal basis of $L^{2}(Z)$ consisting of Maass forms. The method is based on holomorphic extension of irreducible representations of $G$\ud to a certain $G$-invariant domain in $X_{\mathbb{C}}:=G_{\mathbb{C}}/If_{\mathbb{C}}$ (we assume that $G\subset G_{\mathbb{C}})$ .\ud In [2] the method was applied in the case of $G=SL_{2}(\mathbb{R})$ . The method was carried further by Kr\"otz and Stanton in [9], where the results of [2] were slightly improved for $G=SL_{2}(\mathbb{R})$ , and similar results\ud for other rank 1 Riemannian symmetric spaces $G/K$ were obtained. In addition some higher rank cases were considered in [9]. These considerations gave rise to various interesting issues concerning holomorphic\ud extensions of representations and their matrix coefficients