A uniform classification of discrete series representations of affine Hecke algebras

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{H}_{\mathbf{v}}$, in terms of a canonically defined basis $\mathcal{B}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{H}$, and to all $\mathbf{v}\in\mathcal{Q}$, where $\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{H}$. By analytic Dirac induction we define for each $b\in \mathcal{B}_{gm}$ a continuous (in the sense of [OS2]) family $\mathcal{Q}^{reg}_b:=\mathcal{Q}_b\backslash\mathcal{Q}_b^{sing}\ni\mathbf{v}\to\operatorname{Ind}_{D}(b;\mathbf{v})$, such that $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$ (for some $\epsilon(b;\mathbf{v})\in\{\pm 1\}$) is an irreducible discrete series character of $\mathcal{H}_{\mathbf{v}}$. Here $\mathcal{Q}^{sing}_b\subset\mathcal{Q}$ is a finite union of hyperplanes in $\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\operatorname{Ind}_{D}(b;\mathbf{v})$ are piecewise rational in the parameters $\mathbf{v}$. Remarkably, the formal degree of $\operatorname{Ind}_{D}(b;\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in \mathcal{B}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon(b;\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.Comment: 31 pages, 2 table

Local Langlands Correspondence for Classical Groups and Affine Hecke Algebras

Using the results of J. Arthur on the representation theory of classical groups with additional work by Colette Moeglin and its relation with representations of affine Hecke algebras established by the author, we show that the category of smooth complex representations of a split $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). A statement of this kind had been conjecture by G. Lusztig. All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context. We get also parameterizations of representations of affine Hecke algebras, which seem not all to be in the literature yet. All this should also shed some light on what is known as the stable Bernstein center.Comment: 29 pages; this is an updated version (among others, the statement of the definition of the inertial orbit of a Langlands parameter W_F->^LG was corrected and the proposition 1.3 improved

Lectures on conformal field theory and Kac-Moody algebras

This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras.Comment: 59 pages, LaTeX2e, extended version of lectures given at the Graduate Course on Conformal Field Theory and Integrable Models (Budapest, August 1996), to appear in Springer Lecture Notes in Physic

Affine Symmetries of Orbit Polytopes

An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again $G$ if $G$ is itself the affine symmetry group of some orbit polytope, or if $G$ is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows. We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math. 222 (2009), 431--452, Conjecture~5.4].Comment: v2: Referee comments implemented, last section updated. Numbering of results changed only in Sections 9 and 10. v3: Some typos corrected. Final version as published. 36 pages, 5 figures (TikZ

Higher level WZW sectors from free fermions

We introduce a gauge group of internal symmetries of an ambient algebra as a new tool for investigating the superselection structure of WZW theories and the representation theory of the corresponding affine Lie algebras. The relevant ambient algebra arises from the description of these conformal field theories in terms of free fermions. As an illustration we analyze in detail the \son\ WZW theories at level two. In this case there is actually a homomorphism from the representation ring of the gauge group to the WZW fusion ring, even though the level-two observable algebra is smaller than the gauge invariant subalgebra of the field algebra.Comment: LaTeX2e, 30 page

On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

Let $\pi$ be an irreducible smooth complex representation of a general linear $p$-adic group and let $\sigma$ be an irreducible complex supercuspidal representation of a classical $p$-adic group of a given type, so that $\pi\otimes\sigma$ is a representation of a standard Levi subgroup of a $p$-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate $p$-adic classical group obtained by (normalized) parabolic induction from $\pi\otimes\sigma$ does not depend on $\sigma$, if $\sigma$ is "separated" from the supercuspidal support of $\pi$. (Here, "separated" means that, for each factor $\rho$ of a representation in the supercuspidal support of $\pi$, the representation parabolically induced from $\rho\otimes\sigma$ is irreducible.) This was conjectured by E. Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set $I$ of inertial orbits of supercuspidal representations of $p$-adic general linear groups, the category \CC _{I,\sigma} of smooth complex finitely generated representations of classical $p$-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by $\sigma$ and $I$, show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type $A$, $B$ and $D$ and establish functoriality properties, relating categories with disjoint $I$'s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato's exotic geometry.Comment: 21 pages, the results of the paper have been improved thanks to the remarks and encouragements of the anonymous refere

On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups

In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G_2.Comment: 17 pages; v2: several minor corrections, references added; v3: corrections in the table with unipotent discrete series of G

A formula of Arthur and affine Hecke algebras

Let $\pi, \pi'$ be tempered representations of an affine Hecke algebra with positive parameters. We study their Euler--Poincar\'e pairing $EP (\pi,\pi')$, the alternating sum of the dimensions of the Ext-groups. We show that $EP (\pi,\pi')$ can be expressed in a simple formula involving an analytic R-group, analogous to a formula of Arthur in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over nonarchimedean local fields of arbitrary characteristic.Comment: 22 page