For any reductive group
G
G
over a global function field, we use the cohomology of
G
G
-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for
G
G
in the “automorphic to Galois” direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.
The Baum-Connes conjecture [BC], [BCH] proposes a formula for the operator K-theory of reduced group C * -algebras and foliation C * -algebras. If G is the fundamental group of a finite CW -complex then the Baum-Connes conjecture for G can be viewed as an analytic counterpart of the Borel conjecture in manifold theory, which proposes a homological formula for the L-theory of the group ring Z[G]. Moreover the Baum-Connes conjecture for a group G actually implies Novikov's conjecture that the higher signatures of a closed, oriented manifold with fundamental group G are oriented homotopy invariants. For this reason manifold theory has been a driving force behind work on the Baum-Connes conjecture, and in return operator K-theory techniques have proved some of the best known results on the homotopy invariance of higher signatures.From the very beginning, generalizations to group actions have played an important role in the development of the Baum-Connes conjecture. More recently, further extensions have been proposed to general locally compact groupoids [T1] and to coarse geometric spaces [HiR], [R]. Current operator algebraic approaches to the Novikov conjecture rely quite heavily on these (see for instance [Hi]).The Baum-Connes conjecture and its generalizations have now been verified in a variety of cases. For recent work on groups and group actions see [HiK], [L]; for work on groupoids see [T1]; and for work on coarse geometric spaces see [Y]. Indeed the scope of what has now been proved is quite remarkable, especially given the scant information which Baum and Connes had available to them at the outset. Of course the general Baum-Connes conjecture is broader still, applying as it does to every (second countable) locally compact groupoid, or even, in the case of the conjecture 'with coefficients', to every action of a such a groupoid on a C * -algebra. The conjecture has fascinating points of contact not only with the Novikov conjecture but with Riemannian geometry, the representation theory of real and p-adic groups, and the spectral theory of discrete groups.
Nous proposons un renforcement de la propriété (T) de Kazhdan en remplaçant les représentations unitaires par les représentations dans des espaces de Hilbert qui ne sont pas nécessairement unitaires, mais à croissance exponentielle suffisamment petite. On sait qu'un groupe localement compact a la propriété (T) si et seulement si il existe un idempotent p dans C * max (G) tel que pour toute représentation unitaire continue (H, π) de G, π(p) est le projecteur orthogonal sur le sous-espace fermé de H formé des vecteurs Ginvariants. Pour la propriété (T) renforcée, nous posons une définition dans ce style. On appelle longueur sur un groupe localement compact G une fonction continue : G → R + vérifiant (g −1) = (g) et (g 1 g 2) ≤ (g 1) + (g 2) pour g, g 1 , g 2 ∈ G. Dans la suite, pour tout groupe localement compact G, on supposera choisie une mesure de Haar à gauche, qui munit C c (G) d'une structure d'algèbre, par convolution. Définition 0.1 Soit G un groupe localement compact. Si est une longueur sur G, on note E G, la classe des représentations continues π de G dans un espace de Hilbert H telles que π(g) L(H) ≤ e (g) pour tout g ∈ G, et on note C (G) l'algèbre de Banach involutive (pour l'involution usuelle) complétion de C c (G) pour la norme f = sup (H,π)∈E G, π(f) L(H) (en particulier pour = 0, on a C 0 (G) = C * max (G)). On dit que G a la propriété (T) renforcée si pour toute longueur sur G, il existe s > 0, tel que pour tout C ∈ R + , il existe un idempotent autoadjoint p dans C s +C (G) tel que pour tout (H, π) ∈ E G,s +C , π(p) ait pour image le sous-espace de H formé des vecteurs G-invariants
For any 1 ≤ p ≤ ∞ different from 2, we give examples of noncommutative L p spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p > 4 or p < 4/3 and r ≥ 3 these examples are the non-commutative L p
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