Searcharxiv⌕ Search

arXiv subjects

Pierre Clare

Publications and source records attributed to Pierre Clare.

14 recordsLinked to original sources
arXiv2026

Reduced C*-algebras and K-theory for reductive $p$-adic groups

We calculate the $K$-theory of the reduced $C^*$-algebra $C^*_r(G)$ of a reductive $p$-adic group $G$. To do so, we show that each direct summand in Plymen's Plancherel decomposition of $C^*_r(G)$ is Morita equivalent to a twisted crossed product for an action of a finite group on the blow-up of a compact torus along the zero-locus of a certain Plancherel density. It follows that the $K$-theory of $C^*_r(G)$ is the direct sum of the twisted equivariant $K$-theory groups of these blow-ups, which can be computed using an Atiyah-Hirzebruch spectral sequence. As an illustration, the case of $\operatorname{Sp}_4$ is treated in some detail. Our main result is obtained from a more general study of $C^*$-algebras of compact operators on twisted equivariant Hilbert modules, from which we also recover results due to Wassermann for real groups, and to Afgoustidis and Aubert in the $p$-adic case, as well as a new description of the tempered dual of $G$ as a topological space.

math.RT↗
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↗
arXiv2025

A Mackey embedding for reduced C*-algebras of real reductive groups

The purpose of this paper is construct an embedding of the C*-algebra of the Cartan motion group of a real reductive group G into the reduced C*-algebra of G itself. The embedding has a number of applications: we shall use it to characterize the Mackey bijection from the tempered dual of G into the unitary dual of the motion group; to characterize the continuous field of reduced group C*-algebras arising from the contraction of G to its Cartan motion group; and to characterize the Connes-Kasparov assembly map in operator K-theory. Our results continue and complete a project that was begun several years ago by the last two authors, who considered the case of complex groups. In the real case, detailed information from the theory of R-groups is used in the construction.

math.RT↗
arXiv2022

On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

This is the first of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we shall give details of the C*-algebraic Morita equivalence, and then compute the Connes-Kasparov morphism subject to some results in tempered representation theory that we shall prove in Part II using tools from David Vogan's classification of the tempered dual.

math.RT↗
arXiv2022

On the Connes-Kasparov isomorphism, II: The Vogan classification of essential components in the tempered dual

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we presented the Morita equivalence and the Connes-Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual.

math.RT↗
arXiv2017

C*-algebraic normalization and Godement-Jacquet factors

We observe how certain distribution on $\mathrm{GL}(n)$ that generates Godement-Jacquet $\gamma$-factors appears naturally in the C*-algebraic normalization process for standard intertwining integrals on $\mathrm{SL}(n+1)$.

math.RT↗
arXiv2016

Adjoint functors between categories of Hilbert C*-modules

Let E be a (right) Hilbert C*-module over a C*-algebra A. If E is equipped with a left action of a second C*-algebra B, then tensor product with E gives rise to a functor from the category of Hilbert B-modules to the category of Hilbert A-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in a previous paper.

math.OA↗
arXiv2014

Parabolic induction and restriction via C*-algebras and Hilbert C*-modules

This paper is about the reduced group C*-algebras of real reductive groups, and about Hilbert C*-modules over these C*-algebras. We shall do three things. First we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced C*-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced C*-algebra to determine the structure of the Hilbert C*-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in the sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction.

math.RT↗
arXiv2013

C*-algebraic intertwiners for principal series: case of SL(2)

We construct and normalise intertwining operators at the level of Hilbert modules describing the principal series of SL(2). Normalisation is achieved through the use of a Fourier transform defined on some homogenous space and twisted by a Weyl element. Normalising factors are also explicitely obtained. In the appendix we relate reducibility points to a certain distribution arising from the non-normalised intertwiners.

math.RT↗
arXiv2012

On the degenerate principal series of complex symplectic groups

We apply techniques introduced by Clerc, Kobayashi, Orsted and Pevzner to study the degenerate principal series of Sp(n,C). An explicit description of the K-types is provided and Knapp-Stein normalised operators are realised a symplectic Fourier transforms, and their K-spectrum explicitely computed. Reducibility phenomena are analysed in terms of K-types and eigenvalues of intertwining operators. We also construct a new model for these representations, in which Knapp-Stein intertwiners take an algebraic form.

math.RT↗
arXiv2011

Hilbert modules associated to parabolically induced representations of semisimple Lie groups

Given a measured space X with commuting actions of two groups G and H satisfying certain conditions, we construct a Hilbert C*(H)-module E(X) equipped with a left action of C*(G), which generalises Rieffel's construction of inducing modules. Considering G to be a semisimple Lie group and H to be the Levi component L of a parabolic subgroup P=LN, the Hilbert module associated to X=G/N encodes the P-series representations of G coming from parabolic subgroups associated to P. We provide several descriptions of this Hilbert module, corresponding to the classical pictures of P-series. We then characterise the bounded operators on E(G/N) that commute to the left action of C*(G) as central multipliers of C*(L) and interpret this result as a globalised generic irreducibility theorem. Finally, we establish the convergence of intertwining integrals on a dense subset of E(G/N).

math.OA↗