arXiv subjects

Eric Opdam

Publications and source records attributed to Eric Opdam.

At least 19 recordsLinked to original sources

arXiv2024

On Harish-Chandra's Isomorphism

This is the text of a talk given by the first author at the Harish-Chandra centenary meeting held in Allahabad in October 2023. It reviews Harish-Chandra's isomorphism and its many applications to representation theory and mathematical physics. It also announces the existence and uniqueness of nonsymmetric shift operators for an arbitrary root system. These are differential-reflection operators with a transmutation property relative to Dunkl-Cherednik operators: they shift the parameter k of these operators by 1, and restrict on symmetric functions to the hypergeometric shift operators introduced by the first author.

math.RT↗

arXiv2024

Memories of Ian G. Macdonald

This is a slightly edited translation of a paper in Dutch which appeared in Nieuw Archief voor Wiskunde (5) 25 (2024), No.2, 87-90 on the occasion of I.G. Macdonald's death in 2023, and aimed at a very broad mathematical audience. First we review some of Macdonald's most important older results. Then we focus on the period 1985-1995 when Macdonald often visited the Netherlands and there was much interaction between his work, notably the Macdonald polynomials, and the work by the authors. We end with some glimpses about Macdonald as a person.

math.HO↗

arXiv2024

A remark on the Langlands correspondence for tori

For an algebraic torus defined over a local (or global) field $F$, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to-one) homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. Finally, we compare the group of unramified characters with the Galois (co)invariants of the dual torus.

math.RT↗

arXiv2023

Hermitian duals and generic representations for affine Hecke algebras

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between parabolic induction/restriction and Hermitian duals, Bernstein's second adjointness and generalizations of the Langlands classification. We check that, in the known cases of equivalences between module categories of affine Hecke algebras and Bernstein blocks for reductive p-adic groups, such equivalences preserve Hermitian duality. We also initiate the study of generic representation of affine Hecke algebras. Based on an analysis of the Hecke algebras associated to generic Bernstein blocks for quasi-split reductive p-adic groups, we propose a fitting definition of genericity for modules over affine Hecke algebras. With that notion we prove special cases of the generalized injectivity conjecture, about generic subquotients of standard modules for affine Hecke algebras.

math.RT↗

arXiv2022

Residue distributions, iterated residues, and the spherical automorphic spectrum

Let $G$ be a split reductive group over a number field $F$. We consider the computation of the inner product of two $K$-spherical pseudo Eisenstein series of $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following a classical approach initiated by Langlands. We show that only the singularities of the intertwining operators due to the poles of the completed Dedekind zeta function $\Lambda_F$ contribute to the spectrum, while the singularities caused by the zeroes of $\Lambda_F$ do not contribute to any of the iterated residues which arise as a result of the necessary contour shifts. In the companion paper [DMHO] we use this result to explicitly determine the spectral measure of $L^2(G(F)\backslash G(\mathbb{A}_F),\xi)^K_{[T,\mathcal{O}(1)]}$ by a comparison of the iterated residues with the residue distributions of [HO1].

math.RT↗

arXiv2019

On formal degrees of unipotent representations

Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation $\pi$ can be expressed in terms of the adjoint $\gamma$-factor of the enhanced L-parameter of $\pi$. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $\gamma$-factors.

math.RT↗

arXiv2018

Affine Hecke algebras and the conjectures of Hiraga, Ichino and Ikeda

Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field $F$, in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if $F$ is $p$-adic and $G$ is of adjoint type and splits over an unramified extension of $F$. This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143--2207]. More generally for $G$ connected reductive (still assumed to be split over an unramified extension of $F$), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op. cit.

math.RT↗

arXiv2018

Supercuspidal unipotent representations: L-packets and formal degrees

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $\gamma$-factors of L-parameters. This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensueing L-packets, in terms of data from the affine Dynkin diagram of G. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

math.RT↗

arXiv2016

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.

math.RT↗

arXiv2015

On the unramified spherical automorphic spectrum

For an unramified connected reductive group $G$ defined over a number field $F$, consider the part of the spherical automorphic spectrum with cuspidal support $[T,\mathcal{O}(\chi)]$, where $T$ is a maximal torus and $\chi$ is an unramified automorphic character. We define a normalization of the Eisenstein series and we give the precise spectral decomposition of the closure of the subspace spanned by the normalized pseudo-Eiseinstein series. The proof uses residue distributions which were introduced by the third author (in joint work with G. Heckman) in the study of graded affine Hecke algebras, which is an ingredient of a purely local nature. In the case when $G$ is split and $\chi$ is the trivial character, we show that the normalized spectrum is in fact the whole spherical automorphic spectrum. The necessary argument to conclude the result in the split case are based on combinatorial results proved in [DMHO].

math.RT↗

arXiv2015

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.

math.RT↗

arXiv2015

On a uniqueness property of cuspidal unipotent representations

The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are $q$ and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this is verified by the first name author in arXiv:1708.09547. In the present paper we treat the classical families. The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in arXiv:1310.7193) between unipotent affine Hecke algebras (up to twisting by unramified characters). In arXiv:1310.7790 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

math.RT↗

arXiv2013

Spectral transfer morphisms for unipotent affine Hecke algebras

In this paper we will give a complete classification of the spectral transfer morphisms between the unipotent affine Hecke algebras of the various inner forms of a given quasi-split absolutely simple algebraic group, defined over a non-archimidean local field $\textbf{k}$ and split over an unramified extension of $\textbf{k}$. As an application of these results, the results of [O4] on the spectral correspondences associated with such morphisms and some results of Ciubotaru, Kato and Kato [CKK] we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on the formal degrees and adjoint gamma factors for all unipotent discrete series characters of unramified simple groups of adjoint type defined over $\bf{k}$.

math.RT↗

arXiv2013

Spectral correspondences for affine Hecke algebras

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations.

math.RT↗

arXiv2013

The central support of the Plancherel measure of an affine Hecke algebra

We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system and a Weyl group invariant real valued parameter function on the root system. The method is based on the role of this shifted root hyperplane arrangement for the harmonic analysis of affine Hecke algebras. In addition this yields a conceptual proof of the description of the central support of the Plancherel measure of an affine Hecke algebra given in math.RT.0101007v4.

math.RT↗

arXiv2012

Resolutions of tempered representations of reductive p-adic groups

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our method is based on the Bruhat-Tits building of G and on analytic properties of the Schneider-Stuhler resolutions.

math.RT↗

arXiv2012

Extensions of tempered representations

Let $π, π'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (π,π')$ explicitly in terms of the representations of analytic R-groups corresponding to $π$ and $π'$. The result has immediate applications to the computation of the Euler-Poincaré pairing $EP(π,π')$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(π,π')$ is equal to Arthur's formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over non-archimedean local fields of arbitrary characteristic. This sheds new light on the formula of Arthur and gives a new proof of Kazhdan's orthogonality conjecture for the Euler-Poincaré pairing of admissible characters.

math.RT↗

arXiv2010

A formula of Arthur and affine Hecke algebras

Let $π, π'$ be tempered representations of an affine Hecke algebra with positive parameters. We study their Euler--Poincaré pairing $EP (π,π')$, the alternating sum of the dimensions of the Ext-groups. We show that $EP (π,π')$ 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.

math.RT↗