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Paul Broussous

Publications and source records attributed to Paul Broussous.

15 recordsLinked to original sources
arXiv2024

A distinction criterion for Iwahori-spherical representations

Let $G/H$ be a Galois symmetric space for an unramified quadratic extension of a locally compact field $F$, where the group $H$ is semisimple, simply connected, defined and split over $F$. We prove that there exists a subgroup $Γ= Γ(G/H)$ of the group of invertible elements of the Iwahori-Hecke algebra $\mathcal H$ of $G$ such that an Iwahori-spherical representation of $G$ is $H$-distinguished if and only if the corresponding Iwahori-Hecke module is "$Γ$-distinguished".

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arXiv2023

On the distinction of Iwahori-spherical representations

Let $E/F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb H$ a simply connected semisimple algebraic group defined and split over $F$. We establish general results (multiplicities, test vectors) on $\HH (F)$-distinguished Iwahori-spherical representations of $\HH (E)$. For discrete series Iwahori-spherical representations of $\HH (E)$, we prove a numerical criterion of $\HH (F)$-distinction. As an application, we classify the $\HH (F)$-distinguished discrete series representations of $\HH (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.

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arXiv2020

Explicit matrix coefficients and test vectors for discrete series representations

For the discrete series representations of ${\rm GL}(n)$ over a non-archimedean local field $F$, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level $0$ case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level $0$ case, for the Galois symmetric space ${\rm GL}(n,F)/{\rm GL}(n,F_0 )$, for any unramified quadratic extension $F/F_0$. We also exhibit explicit test vectors when these representations are distinguished.

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arXiv2019

Multiplicity one for pairs of Prasad--Takloo-Bighash type

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the centralizer of $E^\times$ in $G$. Using a Galois descent argument, we prove that all double cosets $H g H\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo's result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irreducible representations of $G$ are self-dual and this implies that $(G,H)$ is a Gelfand pair: \[dim_{\mathbb{C}}(Hom_{H}(π,\mathbb{C}))\leq 1\] for all smooth irreducible representations $π$ of $G$. Finally we explain how to obtain the the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch, and we then show self-duality of irreducible distinguished representations in the archimedean case too.

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arXiv2018

Branching laws for the Steinberg representation: the rank 1 case

Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,π)$, for any irreducible representation $π$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde π} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of $H$ (here ${\tilde π}$ denote the contragredient representation, and $1$ the trivial character). In other words we show that there is a sort of `reciprocity law' relating two different branching problems.

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arXiv2017

Distinction of representations via Bruhat-Tits buildings of p-adic groups

Introductory and pedagogical treatmeant of the article : P. Broussous "Distinction of the Steinberg representation", with an appendix by François Courtès, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean Morlet, Dipendra Prasad, Volker Heiermann Ed. 2017. Contains modified and simplified proofs of loc. cit. This article is written in memory of François Courtès who passed away in september 2016.

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arXiv2015

Formule de caractère pour la série discrète de GL(N). Guide de l'utilisateur pour typistes

Ces notes en français sont un résumé de la prépublication arXiv:1402.2501, où en collaboration avec Peter Schneider, nous établissons des formules de caractère pour la série discrète de GL(N) d'un corps local non archimédien. Elles s'adressent à un public de spécialistes suffisamment à l'aise avec les notations et les concepts de la Théorie des Types de Bushnell et Kutzko. These notes, written in french, are a summary of the preprint arXiv:1402.2501, where in collaboration with Peter Schneider, we establish character formulas for the discrete series of GL(N) of a non archimedean local field. They are written for specialists who feel sufficiently confortable with the notation and concept of Type Theory as developped by Bushnell and Kutzko.

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arXiv2014

Coefficient systems and Jacquet modules

Let F be a locally compact non-archimedean field and G the group of F-rational points of an algebraic group assumed to be defined over F, semisimple, simply connected and of F-rank 1. Let pi be a complex irreducible supercuspidal representation of G. We prove that pi is "nearly" induced in the following sense. There exist a maximal compact subgroup K of G and an irreducible smooth representation lamba of K such that pi contains lambda by restriction to K and such that the representation compactly induced from lambda to G is a finite direct sum of irreducible supercuspidal representations. The proof relies on the Schneider and Stuhler theory of equivariant coefficient systems and on a lemma on coefficient systems and Jacquet modules.

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arXiv2014

Simple characters and coefficient systems on the building

Let F be a non-archimedean local field and G be the group GL(N,F). Let πbe a smooth complex representation of G lying in the Bernstein block B(π) of some simple type in the sense of Bushnell and Kutzko. Refining the approach of the second author and U. Stuhler, we canonically attach to πa subset X_πof the Bruhat-Tits building X of G, as well as a G-equivariant coefficient system C[π] on X_π. Roughly speaking the coefficient system is obtained by taking isotypic components of πaccording to some representations constructed from the Bushnell and Kutzko type of π. We conjecture that when πhas central character, the augmented chain complex associate to C[π] is a projective resolution of πin the category B(π). Moreover we reduce this conjecture to a technical lemma of representation theoretic nature. We prove this lemma when πis an irreducible discrete series of G. We then attach to any irreducible discrete series πof G an explicit pseudo-coefficient f_πand obtain a Lefschetz type formula for the value of the Harish-Chandra character of πat a regular elliptic element. In contrast to that obtained by U. Stuhler and the second author, this formula allows explicit character value computations.

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arXiv2012

Distinction of the Steinberg representation

We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.

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arXiv2012

Transfert du pseudo-coefficient de Kottwitz et formules de caractere pour la serie discrete de GL(N) d'un corps local

Let G be the group GL(N,F), where F is a non-archimedean locally compact field. Using Bushnell and Kutzko's simple types, as well as an original idea of Henniart's, we construct explicit pseudo-coefficients for the discrete series representations of G. As an application we deduce new formulas for the value of the Harish-Chandra character of certain such representations at certain elliptic regular elements.

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arXiv2010

Smooth representations of GL(m,D), V: Endo-classes

Let F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of GL(n) over F the notion of endo-class introduced by Bushnell and Henniart for GL(n,F). We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of GL(n,F) an endo-class over F. We conjecture that this endo-class is invariant under the local Jacquet-Langlands correspondence.

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arXiv2007

Representations of PGL(2) of a local field and harmonic forms on simplicial complexes

We give combinatorial models for complex, smooth, non-spherical, generic, irreducible representations of the group G=PGL(2,F), where F is a non-archimedean locally compact field. They use the graphs X_k lying above the tree of G, introduced in a previous work. We show that such representations may be realized as quotients of the cohomology of X_k for some k, or equivalently as spaces of discrete harmonic forms on X_k. For supercuspidal representations these models are unique.

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arXiv2004

Acyclicity of Schneider and Stuhler's coefficient systems:another approach in the level 0 case

Let F be a non archimedean local field and G be the locally profinite group GL(N,F), N>0. We denote by X the Bruhat-Tits building of G. For all smooth complex representation V of G and for all level n>0, Schneider and Stuhler have constructed a coefficient system C = C(V, n) on the simplicial complex X. They proved that if V is generated by its fixed vectors under the principal congruence subgroup of level n, then the augmented complex of oriented chains of X with coefficients in C is a resolution of V in the category of smooth complex representations of G. In this paper we give another proof of this result, in the level 0 case, and assuming moreover that V is generated by its fixed vectors under an Iwahori subgroup I of G. Here "level 0" refers to Bushnell and Kutzko's terminology, that is to the case n=1+0. Our approach is different. We strongly use the fact that the trivial character of I is a type in the sense of Bushnell and Kutzko.

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