Searcharxiv⌕ Search

arXiv subjects

Volker Heiermann

Publications and source records attributed to Volker Heiermann.

18 recordsLinked to original sources
arXiv2025

Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups

For a finite central extension $\tilde{G}$ of a classical $p$-adic reductive group, we consider the endomorphism algebra of some induced projective generator \`a la Bernstein of the category of smooth representations of $\tilde{G}$. In the case where the Levi subgroups decompose, we compute this algebra to get a result similar to the one previously obtained by the first author for classical $p$-adic groups, showing that this intertwining algebra is a twisted semi-direct product of an affine Hecke algebra with parameters by a twisted finite group algebra. We discuss also the case with non-decomposed Levi subgroups. We then give an application to the category of genuine representations of a $p$-adic metaplectic group. Using results of C. M\oe glin relative to the Howe correspondence, we show that the Bernstein components of these groups are equivalent to tensor products of categories of unipotent representations of classical groups. This generalizes a previous result of the first author. It implies an equivalence of categories between the category of genuine representations of the $p$-adic metaplectic group and the direct sums of those of smooth representations of the corresponding split odd special orthogonal group and its pure inner form.

math.RT↗
arXiv2022

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 $[T,\mathcal{O}(χ)]$, where $T$ is a maximal torus and $χ$ 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 $χ$ 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↗
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 $Λ_F$ contribute to the spectrum, while the singularities caused by the zeroes of $Λ_F$ 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 $L^2(G(F)\backslash G(\mathbb{A}_F),ξ)^K_{[T,\mathcal{O}(1)]}$ by a comparison of the iterated residues with the residue distributions of [HO1].

math.RT↗
arXiv2018

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

Let $π$ be an irreducible smooth complex representation of a general linear $p$-adic group and let $σ$ be an irreducible complex supercuspidal representation of a classical $p$-adic group of a given type, so that $π\otimesσ$ 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 $π\otimesσ$ does not depend on $σ$, if $σ$ is "separated" from the supercuspidal support of $π$. (Here, "separated" means that, for each factor $ρ$ of a representation in the supercuspidal support of $π$, the representation parabolically induced from $ρ\otimesσ$ is irreducible.) This was conjectured by E. Lapid and M. Tadić. (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 $\CC _{I,σ}$ of smooth complex finitely generated representations of classical $p$-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by $σ$ 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.

math.RT↗
arXiv2018

The value of the global intertwining operators on spherical vectors

Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.

math.NT↗
arXiv2016

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.

math.RT↗
arXiv2016

A note on Standard Modules and Vogan L-packets

Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(τ,ν)$ and $I(τ',ν')$ for $G$ and $G'$ respectively with $τ'$ a generic tempered representation, such that the Harish-Chandra's $μ$-functions of a representation in the supercuspidal support of $τ$ and of a generic essentially square-integral representation in some Jacquet module of $τ'$ agree (after a suitable identification of the underlying spaces under which $ν=ν'$), we show that $I(τ,ν)$ is irreducible whenever $I(τ',ν')$ is. The conditions are satisfied if the Langlands quotients $J(τ,ν)$ and $J(τ',ν')$ of respectively $I(τ,ν)$ and $I(τ',ν')$ lie in the same Vogan $L$-packet (whenever this Vogan $L$-packet is defined), proving that, for any Vogan $L$-packet, all the standard modules whose Langlands quotient is equal to a member of the Vogan $L$-packet are irreducible, if and only if this Vogan $L$-packet contains a generic representation. The result for generic Vogan $L$-packets of quasi-split orthogonal and symplectic groups was proven by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these Groups.

math.NT↗
arXiv2015

On the generic local Langlands correspondence for GSpin groups

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of $GSpin$, by constructing Langlands parameters for general generic irreducible admissible representations of $GSpin $ from the one for generic irreducible supercuspidal representations of its Levi subgroups.

math.NT↗
arXiv2011

Algèbres de Hecke avec paramètres et représentations d'un groupe p-adique classique: préservation du spectre tempéré

Let G be an orthogonal or symplectic p-adic group (not necessarily split) or an inner form of a general linear p-adic group. In a previous paper, it was shown that the Bernstein components of the category of smooth representations of G are equivalent to the category of right modules over some Hecke algebra with parameters, or more general over the semi-direct product of such an algebra with a finite group algebra. The aim of the present paper is to show that this equivalence preserves the tem- pered spectrum and the discrete series representations.

math.RT↗
arXiv2009

On the tempered L-function conjecture

We give a general proof of Shahidi's tempered L-function conjecture, which has previously been known in all but one case. One of the consequences is the standard modules conjecture for p-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a p-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.

math.NT↗
arXiv2009

Paramètres de Langlands et Algèbres d'entrelacement

Let G be a classical p-adic group and $(ψ,ε)$ the Langlands parameter of an irreducible supercuspidal representation of a Levi subgroup of G. Using data from $(ψ,ε)$, we determine explicitly the intertwining algebra of the representation which is induced from the orbit of the supercuspidal representation associated to $(ψ,ε)$.

math.RT↗
arXiv2008

Op\'erateurs d'entrelacement et alg\`ebres de Hecke avec param\`etres d'un groupe r\'eductif $p$-adique - le cas des groupes classiques

For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators \`a la Bernstein of the category of smooth representations of $G$ and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected $p$-adic group.

math.RT↗
arXiv2005

Unipotent orbits and local L-functions

The result in theorem 2.1 has been strengthened (see theorem 2.3) and the remarks in the introduction and the text adapted to this new result. Also some misprints in the previous version have been corrected.

math.NT↗
arXiv2004

Orbites unipotentes et poles d'ordre maximal de la fonction $μ$ de Harish-Chandra

In a previous work, we have shown that a representation of a $p$-adic group obtained by (normalized) parabolic induction from an irreducible supercuspidal representation $σ$ of a Levi subgroup $M$ contains a subquotient which is square integrable, if and only if Harish-Chandra's $μ$-function has a pole in $σ$ of order equal to the parabolic rank of $M$. The aim of the present article is to interpret this result in terms of Langlands' functoriality principle.

math.RT↗
arXiv2003

Decomposition spectrale et representations speciales d'un groupe reductif p-adique

Let G be a reductive connected p-adic group. With help of the Fourier inversion formula used in [Une formule de Plancherel pour l'algebre de Hecke d'un groupe reductif p-adique - V. Heiermann, Comm. Math. Helv. 76, 388-415, 2001] we give a spectral decomposition on G. In particular we deduce from it essentially that a cuspidal representation of a Levi subgroup M is in the cuspidal support of a square integrable representation of G, if and only if it is a pole of Harish-Chandra's μ-function of order equal to the parabolic rank of M. This result has been conjectured by A. Silberger in 1978. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra's μ-function which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coeffieicients. These identities contain informations on the formel degree of square integrable representations and on their position in the induced representation.

math.RT↗