Nombres de Tamagawa des groupes réductifs quasi-connexes

We define Tamagawa numbers for quasi-connected linear algebraic groups. This category of non-connected linear algebraic groups occurs naturally in the stabilization of the Arthur-Selberg trace formula. An expression for their Tamagawa numbers, needed for the stabilization process, is given in term of cohomological invariants generalizing results of Ono, Sansuc, Kottwitz and Kottwitz & Shestad. Received: 7 March 2000     

Résolutions flasques des groupes réductifs connexes

A connected reductive group G over a (characteristic zero) field k may be written as a quotient $H/S$, where the k-group H is an extension of a quasitrivial torus by a simply connected group, and S is a flasque k-torus, central in H. Such presentations $G=H/S$ lead to a simplified approach to the Galois cohomology of G and related objects, such as the Brauer group of a smooth compactification of G. When k is a number field, one also recovers known formulas, in terms of S, for the quotient of the group $G(k)$ of rational points by R-equivalence, and for the Abelian groups which measure the lack of weak approximation and the failure of the Hasse principle for principal homogeneous spaces. To cite this article: J.-L. Colliot-Thélène, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

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.     

Sur la détermination des structures réelles de groupes simples,finis et continus,au moyen des isomorphies involutives

Sans résumé     

Séries de poincaré des groupes géométriquement finis

In this paper, we study the behaviour of the Poincaré series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

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Durant la rédaction de cet article, M. Peigné a bénéficié d'un détachement au Centre National de la Recherche Scientifique, URA 305.     

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     

Sur une décomposition d'ensembles

Sans résumé     

Quelques remarques sur les actions analytiques des réseaux des groupes de Lie de rang supérieur

All the actions considered here are (real) analytic. Let г be a subgroup of finite index of SL(n,). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus T^a (n ≥ 3), and its standard projective action on the sphere Sⁿ⁻¹ (n, ≥ 4).     

Cohomologie des groupes linéaires,K-théorie de Milnor et groupes de Witt

We construct some remarkable classes in the cohomology of linear groups.     

Une application de la théorie des excursions à une diffusion réfléchie dégénérée

Résumé Nous considérons l'équation différentielle stochastique réfléchie à valeur dansR ² associée à l'opérateur différentiel dégénéré (1/2)²/x ²+x/y. On étudie alors le phénomène d'excursion autour du point origine. En particulier, on peut identifier la loi de l'inverse du temps local comme un subordinateur stable d'ordre 1/4 et en donner une formule du type temps d'occupation. Ceci permet alors d'étudier le comportement de la loi du processus au voisinage du point origine pour préciser, dans ce cas particulier, des résultats obtenus par Krée (1985) grâce à des méthodes analytiques.

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Summary We consider theR ²-valued reflected stochastic differential equation associated with the degenerated differential operator (1/2)²/x ²+x/y. We study the excursion phenomenon around the point (0, 0). We are able to identify the law of the inverse of the local time as a stable subordinator with order 1/4 and to give for it an occupation time formula. These facts enable us to study the behavior of the law of the process near the point (0, 0) and to refine, in our case, results obtained by Krée (1985) by means of analytical methods.

     

Sur une décomposition topologique des fonctions de Sobolev et sur la localisation de l'énergie dans les plasmas idéaux

Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H₀¹ (Ω) of all functions ψ₀ ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ₀ a class W^weak (ψ₀) of functions which captures robust topological properties, of the level sets of {ψ₀} and, moreover, is closed under weak limits.     

Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe

Let X be a smooth connected compact projective variety over . We study the Shafarevich conjecture concerning holomorphic convexity along the lines of Kollárs approach, when the fundamental group of X admits large finite dimensionnal representations. We prove that, given n, the topological covering space of a projective algebraic compact complex manifold X corresponding to the intersection of the kernels of all linear reductive representations of ₁(X) to GL_n() is holomorphically convex. In the surface case, this is a corollary of a theorem due to Katzarkov and Ramachandran.