La conjecture de Baum–Connes pour les feuilletages moyennables

We show, using the construction of Higson and Kasparov, that the Baum–Connes Conjecture holds for foliations whose holonomy groupoid is Hausdorff and amenable. More generally, for every locally compact, -compact and Hausdorff groupoid G acting continuously and isometrically on a continuous field of affine Euclidean spaces, the Baum–Connes conjecture with coefficients is an isomorphism, and G amenable in K-theory. In addition, we show that C^*(G) satisfies the Universal Coefficient Theorem.     

Extension des scalaires par le morphisme de Frobenius,pour les groupes réductifs

Let G be a reductive group over a field k of characteristic $p>0$. For $n?0$ and $q:=p^n$, let $G^{\{n\}}$ be deduced from G by the extension of scalars $x?x^q:k?k$. If k is perfect, this keeps making sense for $n?\mathbb{Z}$. We show that, if k is perfect, there exists $m>0$ such that the algebraic groups G and $G^{\{m\}}$ over k are isomorphic. The isomorphism class of $G^{\{n\}}$, as a reductive group over k, then depends only on n modulo m. For k not necessarily perfect, we show that such a periodicity remains true for n large enough.     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Une correspondance de modules pour les blocs à groupes de défaut abéliens

Let b be a block of a finite group G with an abelian defect group P and an inertial quotient E. Let us denote by L the semi-direct product of P and E. If E is cyclic and acts freely on P−{1}, we prove that the stable categories of Gb- and L-modules are equivalent, as a consequence of a more general result, without any hypothesis on E, about partial covering exomorphisms relating Gb with a suitable twisted group algebra _*

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à Jacques Tits à l'aube de sa sexagénie     

La transformée en ridelettes pour les fonctionnelles de Wiener

The ridgelet transform is extended to Wiener functionals. Even if the reconstruction property is not satisfied, we still approximate a Wiener functional with the help of its transform and we give an application to its regularity.     

Le problème des diviseurs de zéro pour les groupes de Lie nilpotents

Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist $0\ne \alpha \in C_c(G)$ and $0\ne \xi \in L^2(G)$ such that $\alpha 1\xi =0$, contrary to what happens for the group ${\mathbb{R}}^n$. Moreover, the set of zero divisors is a total subset of $L^2(G)$. This result is first proven for the Heisenberg group $H_n$ where it is based on the existence of non-trivial Schwartz functions f satisfying $f1(X_k+\mathrm{i}{Y}_k)=0$ for $1?k?n$. To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Estimations pour les chaînes de Markov réversibles

We give Gaussian lower and upper bounds for reversible Markov chains on a graph under two geometric assumptions (volume regularity and Poincaré inequality). This is first proved for continuous-time Markov chains via a parabolic Harnack inequality. Then, the estimates for the discrete-time Markov chains are derived by comparison.     

Un résultat de densité pour les équations de Maxwell

We prove the density of regular fielclx in the space of square-integrahle fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electmmagnetism equations.     

Stabilité de la conjecture de Baum—Connes pour certains produits semi-directs de groupes

Let Γ = G ⋉_r N be the semi-direct product of two locally compact groups, and A a Y-algebra. We build a map from topological K-theory of Γ with coefficients in A to the one for G with coefficients in N ⋉_r A. For Γ a p-adic group equipped with a γ element and N an amenable group, we prove that Γ satisfies Baum-Connes conjecture with coefficients when G does.     

« Trivialité » du problème de Kashiwara–Vergne pour les algèbres de Lie résolubles

In a recent note Alekseev and Torossian (2009) [4] give a simple algebraic solution of the Kashiwara–Vergne problem for quadratic Lie algebras. Their method also leads to a simple solution for solvable Lie algebras, as shown here.     

Processus de Lévy autosimilaires sur les groupes de Lie

In this Note we study the global autosimilarity property of Lévy processes on Lie groups.     

Dualité pour les groupes de type multiplicatif sur certains corps de fonctions

Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate–Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate–Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K).