arXiv subjects

Alexandre Afgoustidis

Publications and source records attributed to Alexandre Afgoustidis.

12 recordsLinked to original sources

arXiv2026

The Mackey bijection as a stratified equivalence

This paper is about the Mackey analogy between the tempered representation theory of a real reductive group and that of its Cartan motion group. We consider the embedding of reduced C*-algebras constructed recently in connection with the Mackey bijection, and study its behavior on certain natural stratifications of the tempered duals. We formulate our result using a notion of stratified equivalence inspired by the study of the smooth dual of $p$-adic groups via the structure of Hecke algebras, in particular by the work of Aubert, Baum, Plymen and Solleveld. We derive related new topological properties of the Mackey bijection. We also analyze the behavior of the Mackey embedding on a stratification of reduced C*-algebras attached to a partition of the tempered dual into particularly elementary pieces, introduced in recent work of Bradd, Higson and Yuncken.

math.RT↗

arXiv2024

Nilpotent Invariants for Generic Discrete Series of Real Groups

Let $G(\mathbb{R})$ be a real reductive group. Suppose $\pi$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $\pi$ related to nilpotents elements of the Lie algebra of $G$ (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi$ has a Whittaker model. If $\pi$ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related results, including an elementary proof that passage from $\pi$ to the three invariants defines natural bijections between the generic discrete series in an $L$-packet, the possible Whittaker data for $G(\mathbb{R})$, and the appropriate sets of nilpotent orbits. Given one of the three invariants, we also explain how to reconstruct the other two. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant-Sekiguchi correspondence.

math.RT↗

arXiv2024

Roger Godement et les fonctions de type positif

Ce texte, \'ecrit pour la Gazette de la Soci\'et\'e math\'ematique de France, \'evoque les fonctions de type positif et leur histoire avant 1950 ; on y pr\'esente notamment des extraits de lettres \'ecrites par Roger Godement, qui leur consacra sa th\`ese en 1946. This is an expository paper on the early history of positive-definite functions, written for the "Gazette de la Soci\'et\'e Math\'ematique de France". It contains pictures of letters written by Roger Godement, during and after the preparation of his 1946 thesis about positive-definite functions on groups.

math.HO↗

arXiv2024

Lowest $K$-types in the local Langlands correspondence

Consider the irreducible representations of a real reductive group $G(\mathbb{R})$, and their parametrization by the local Langlands correspondence. We ask: does the parametrization give easily accessible information on the restriction of representations to a maximal compact subgroup $K(\mathbb{R})$ of $G(\mathbb{R})$? We find a natural connection between the set of lowest $K$-types of a representation and its Langlands parameters. For our results, it is crucial to use the refined version of the local Langlands correspondence, involving (coverings of) component groups attached to $L$-homomorphisms. The first part of the paper is a simplified description of this refined parametrization.

math.RT↗

arXiv2023

Progrès récents sur les représentations supercuspidales

Let $G$ be a reductive group over a nonarchimedean local field $F$. In the quest for a classification of irreducible smooth representations of $G$, it is critical to understand the case of supercuspidal representations -- those whose matrix coefficients are compactly supported modulo the center. Progress in understanding these representations has been continuous over the past fifty years. In "tame" cases where the residual characteristic of $F$ is big enough for $G$, J.-K. Yu described in 2001 a general construction of supercuspidal representations, building on a large body of work. But recent developments have made the general picture much more complete and much clearer. For instance, the work of J. Fintzen, T. Kaletha and L. Spice provides (in the tame case) a classification of supercuspidal representations, an explicit formula for "almost all" their characters, and an explicit construction of a local Langlands correspondence for supercuspidal $L$-packets. While the basic constructions involve Bruhat--Tits buildings and representations of finite groups, the resulting character formulas and the description of $L$-packets have striking parallels with the case of real groups

math.RT↗

arXiv2022

Invariant Gaussian fields on homogeneous spaces: explicit constructions and mean nodal volume

We review and study some of the properties of smooth Gaussian random fields defined on a homogeneous space, under the assumption that the probability distribution is invariant under the isometry group of the space. We first give an exposition, building on early results of Yaglom, of the way in which representation theory and the associated special functions make it possible to give completely explicit descriptions of these fields in many cases of interest. We then turn to the expected size of the zero-set: extending two-dimensional results from Optics and Neuroscience, we show that every invariant field comes with a natural unit of volume (defined in terms of the geometrical redundancies in the field) with respect to which the average size of the zero-set is given by a universal constant depending only on the dimension of the source and target spaces, and not on the precise symmetry exhibited by the field.

math.PR↗

arXiv2021

On the analogy between real reductive groups and Cartan motion groups. I: The Mackey-Higson bijection

George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspondence between the admissible duals.

math.RT↗

arXiv2021

$C^\ast$-blocks and crossed products for classical $p$-adic groups

Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermann's theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We then focus on the case where $G$ is a quasi-split symplectic, orthogonal or unitary group, and explicitly determine the connected components for which the geometric assumption is satisfied.

math.RT↗

arXiv2020

Continuity of the Mackey-Higson bijection

When $G$ is a real reductive group and $G_0$ is its Cartan motion group, the Mackey-Higson bijection is a natural one-to-one correspondence between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. In this short note, we collect some known facts about the topology of the tempered dual $\widetilde{G}$ and that of the unitary dual $\widehat{G_0}$, then verify that the Mackey-Higson bijection $\widetilde{G} \to \widehat{G_0}$ is continuous.

math.RT↗

arXiv2019

On the analogy between real reductive groups and Cartan motion groups. II: Contraction of irreducible tempered representations

Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $π$ of G, we build, in an appropriate Fréchet space, a family of subspaces and evolution operators that contract $π$ onto the corresponding representation of $G_0$.

math.RT↗

arXiv2019

On the analogy between real reductive groups and Cartan motion groups. III: A proof of the Connes-Kasparov isomorphism

Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov "conjecture"' for the K-theory of reduced groupe $C^\ast$-algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a reductive Lie group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism.

math.OA↗

arXiv2016

Un effet de moiré sur les espaces symétriques de type non-compact

We prove that if $X$ is a symmetric space of the noncompact type, just as adding Helgason waves which propagate in all direction yields an elementary spherical function for $X$, a Helgason wave can be produced by adding elementary spherical functions whose centers cluster along a horocycle in $X$.

math.GR↗