Mémoires de la Société mathématique de France | 2026
Année : 2026
Tome : 188
ISBN : 978-2-37905-228-6
We consider the restriction and induction of representations between a covering group and subgroups closely related to its derived subgroup, both on the representation-theoretic side and the L-parameter side. In particular, restriction of a genuine principal series is analyzed in detail. We also discuss a metaplectic tensor product construction for covers of the symplectic similitudes groups, and remark on the generality of such a construction for other groups. Furthermore, working with an irreducible constituent of a unitary unramified principal series, we prove a multiplicity formula for its restriction to the derived subgroup in terms of three associated R-groups. Later in the paper, we study how the parametrization of elements inside an unramified L-packet varies along with different choices of hyperspecial maximal compact subgroups and their splittings. We also investigate the genericity of elements inside such an L-packet with respect to varying Whittaker datum. Pertaining to the above two problems, covers of symplectic similitudes groups are discussed in detail in the last part of the paper.
Covering groups, L-group, parameters, Whittaker functionals, local coefficients matrix, R-group, unramified representation, L-packet, metaplectic tensor product
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Cours Spécialisés | 2026
Année : 2026
Tome : 33
ISBN : 978-2-37905-225-5
This book discusses the interactions between the (nonlinear) metric structure of Banach spaces and their linear asymptotic behavior. The overarching problem is to understand how the various linear structures of a Banach space are preserved under certain nonlinear maps. The first chapters contain what are by now classical results to study the most basic and fundamental rigidity problems: the Lipschitz or uniform classification of Banach spaces. The other chapters form the main contribution of this book. The intended goal is to cover the work of many researchers, in particular their discoveries from the past 25 years, trying to understand how asymptotic properties of Banach spaces are preserved under several essential notions of nonlinear (bi-Lipschitz, coarse-Lipschitz, coarse or uniform) embeddings. This is part of a broader program called the Kalton program. This program, inspired by the Ribe program, seeks to uncover purely metric characterizations of asymptotic properties of Banach spaces. Many of these charaterizations are closely connected to the geometry of families of metric graphs (trees, Hamming graphs, diamond graphs, interlacing graphs) thus this book is also about the geometric structure of those graphs.
Asymptotic uniform convexity, asymptotic uniform smoothness, asymptotic uniform flatness, Rolewicz property (β)
, coarse embeddings, uniform embeddings, coarse-Lipschitz embeddings, Hamming graphs, Johnson graphs, interlacing graphs, diamond graphs, asymptotic properties of Banach spaces, trees and branches in Banach spaces, Szlenk index, Kalton program, nonlinear rigidity of Banach spaces, universal Banach spaces, Radon-Nikodym property, Lipschitz-free spaces, approximate midpoints principle, Gorelik principle
Astérisque | 2026
Année : 2026
Tome : 463
Nb. de pages : 229
ISBN : 978-2-37905-226-2
For a sufficiently rich group G
of homeomorphisms on the line, we describe the structure of the other possible actions of G
on the line, studying how much they can differ from the standard action. The main assumption is that G
is locally moving, and one such example is given by Thompson’s group F
. We show that in the C1
setting the standard action is essentially the only possible faithful one, whereas the group can admit many exotic C0
actions. Nonetheless, all exotic actions preserve a lamination, and the dynamics on the lamination is reminiscent of the standard action. Out of this we deduce that a great variety of locally moving actions are structurally stable.
Group actions on the real line, locally moving groups, actions on real trees, local rigidity, left-orderable groups, groups of piecewise linear homeomorphisms
Astérisque | 2026
Année : 2026
Tome : 464
Nb. de pages : 407
ISBN : 978-2-37905-227-9
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine L2
-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Mœglin--Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.
Arthur-Selberg trace formula, stable trace formula, metaplectic group