Let $X=G/H$ be a reductive symmetric space and $K$ a maximal compact subgroup of $G$. We study Fourier transforms of compactly supported $K$-finite distributions on $X$ and characterize the image of the space of such distributions.

Soit $G$ un groupe algébrique réductif sur la clôture algébrique d’un corps fini $\mathbb{F}_q$ et défini sur ce dernier. L’existence du support unipotent d’un caractère irréductible du groupe fini $G(\mathbb{F}_q)$, ou d’un faisceau caractère de $G$, a été établie dans différents cas par Lusztig, Geck et Malle, et le second auteur. Dans le présent article, nous démontrons que toute classe unipotente sur laquelle la restriction du caractère ou du faisceau caractère donné est non nulle est contenue l’adhérence de Zariski du support unipotent de ce dernier. Pour établir ce résultat, nous étudions certaines représentations des groupes de Weyl, dites “bien supportées”.

Let $G$ be a reductive algebraic group over the algebraic closure of a finite field $\mathbb{F}_q$, and defined over the latter. The existence of a unipotent support for an irreducible character of the finite group $G(\mathbb{F}_q)$, or for a character sheaf on $G$, has been established in several cases by Lusztig, Geck and Malle, and the second author. In this paper, we prove that any unipotent class on which the restriction of the given character or character sheaf does not vanish is contained in the Zariski closure of its unipotent support. In order to establish this result, we study some representations of Weyl groups, which are called ‘well supported’.

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soulé for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

Dans ce papier nous étudions une correspondance de Jacquet–Langlands locale pour toutes les représentations lisses irréductibles. La correspondance est caractérisée par le fait qu’elle respecte la correspondance de Jacquet–Langlands classique et commute avec le foncteur d’induction parabolique. Elle est compatible dans un sens à préciser au foncteur de Jacquet et à l’involution d’Aubert–Schneider–Stuhler. Nous utilisons cette correspondance pour montrer qu’une certaine classe de représentations d’une forme intérieure de $\mathrm{GL}_n$ sur un corps $p$-adique sont unitarisables. C’est le premier pas dans la preuve de la conjecture U1 de Tadić.

We study a local Jacquet–Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet–Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet’s functor and the involution of Aubert–Schneider–Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet–Langlands correspondence and we deduce that a certain class of representations of an inner form of $\mathrm{GL}_n$ over a $p$-adic field are unitarizable. This is the first step towards the proof of Conjecture U1 of Tadić.

Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_{\mathrm{w}}G(Y)$, where $*_{\mathrm{w}}$ is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula $\mu(G*_{\mathrm{w}}H)=\mu(G)\boxtimes\mu(H)$, where $\mu$ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.

The theory of generalized elliptic curves gives a moduli-theoretic compactification for modular curves when the level is a unit on the base, and the theory of Drinfeld structures on elliptic curves provides moduli schemes over the integers without a modular interpretation of the cusps. To unify these viewpoints it is natural to consider Drinfeld structures on generalized elliptic curves, but some of these resulting moduli problems have non-étale automorphism groups and so cannot be Deligne–Mumford stacks. Artin’s method as used in the work of Deligne and Rapoport rests on a technique of passage to irreducible fibers (where the geometry determines the group theory), and this does not work in the presence of non-étale level structures and non-étale automorphism groups. By making more efficient use of the group theory to bypass these difficulties, we prove that the standard moduli problems for Drinfeld structures on generalized elliptic curves are proper Artin stacks. We also analyze the local structure on these stacks and give some applications to Hecke correspondences.

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic. The main goal of this paper is to introduce a Riemann–Hilbert problem whose solution gives the solution of our initial–boundary-value problem. This is a preliminary step to obtain uniform long-time asymptotics for the solution of this equation using both the stationary phase and the Deift–Zhou methods.

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal{R}$ on a standard Borel space is amenable if and only if the full group $[\mathcal{R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.

Nous construisons une $KK$-théorie pour les algèbres de Banach équivariante par l’action d’un groupoïde et nous montrons la conjecture de Baum–Connes à coefficients commutatifs pour les groupes hyperboliques et pour les groupoïdes de Poincaré des feuilletages à base compacte qui peuvent être munis d’une métrique riemannienne longitudinale à courbure sectionnelle strictement négative.

We construct a $KK$-theory for Banach algebras, equivariant with respect to the action of a groupoid. We prove the Baum–Connes conjecture with commutative coefficients for hyperbolic groups and for the Poincaré groupoids of foliations with a compact base and a longitudinal Riemannian metrics with negative sectional curvature.

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $\mathbb{R}^{n+1}$, for $n\geq3$, which generalize the well known two-dimensional ‘Riemann minimal surfaces’. The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the two-dimensional case, they are not foliated by spheres.

Résumé Nous prouvons l’existence d’une famille à un paramètre d’hypersurfaces de $\mathbb{R}^{n+1}$, pour $n\geq 3$, qui sont minimales et qui généralisent les surfaces minimales de Riemann. Les hypersurfaces que nous obtenons sont des hypersurfaces complètes, simplement périodiques et qui ont une infinité de bouts hyperplans parallèles. Contrairement au cas des surfaces, i.e. $n=2$, ces hypersurfaces ne sont pas feuilletées par des sphères.

Nous présentons une étude des relations d’équivalence mesurées du point de vue spectral. En particulier nous montrons que pour une relation d’équivalence donnée, la propriété T de Kazhdan, l’ergodicité forte et la moyennabilité peuvent être caractérisées par la présence d’un trou dans le spectre de certaines marches aléatoires sur les orbites de cette relation (à coefficients dans des représentations hilbertiennes appropriées). Les démonstrations reposent de façon essentielle sur l’étude des relations d’équivalence moyennables (Connes–Feldman–Weiss), sur les caractérisations spectrales correspondantes, pour les groupes dénombrables, de la moyennabilité (théorème de Kesten) et de la propriété T de Kazhdan (Gromov, Ghys, etc.), ainsi que sur des résultats techniques sur les représentations unitaires de relations d’équivalence mesurées (en particulier au voisinage de la représentation triviale) que nous développons au cours de l’article. Enfin nous obtenons un analogue du “critère $\lambda_1>1/2$” pour les relations d’équivalence mesurées.

It is a well known theorem due to Kesten that amenability for discrete groups can be characterized in terms of the spectra of diffusion operators associated to random walks on the Cayley graph of these groups. In this paper we are interested in analogous results in the framework of discrete measured equivalence relations. Our main results concern characterizations of Kazhdan’s property T, amenability, and the non existence of amenable quotients (strong ergodicity), in terms of the spectra of diffusion operators associated to random walks and hilbertian representations of the underlying equivalence relation. Our arguments are based on the proof of Connes–Feldman–Weiss’s classification of amenable equivalence relations, the spectral characterizations, in the group case, of amenability (Kesten) and property T (Gromov, Ghys, etc.), as well as new results on the representation theory of measured equivalence relations (in particular in the neighbourhood of the trivial representation) and Kazhdan’s property T. As an application we show how Żuk’s ‘$\lambda_1>1/2$’ criterion for property T can be adapted to measured equivalence relations.

In this paper we study the category of non-degenerate modules over the Harish-Chandra Schwartz algebra of a p-adic connected reductive group. We construct functors of parabolic induction and restriction and show that they are exact and both ways adjoint to each other.

Nous considérons l’intersection d’une sous-variété $X$ d’une variété abélienne $A$ avec l’union de tous les sous-groupes de $A$ obtenus comme somme d’un sous-groupe donné $\varGamma$ de rang fini et d’un sous-groupe algébrique de dimension donnée. Nous montrons que, quitte à retirer à $X$ un certain ensemble exceptionnel, l’intersection est de hauteur bornée. L’énoncé est optimal pour une courbe $X$. Nous étudions également les épaississements $\varGamma_{\varepsilon}$ introduits par Poonen. La démonstration repose sur une généralisation uniforme de la méthode de Vojta et sur des calculs de nombres d’intersection de cycles réels sur $A$.

We study the intersection of a subvariety $X$ of an abelian variety $A$ over $\bar{\mathbb{Q}}$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $\varGamma$ and an algebraic subgroup of $A$ of given dimension $d$. Our main result asserts that if we remove a suitable exceptional subset from $X$ then the intersection is a set of bounded height. In some cases, this combines with the output of Part I to yield finiteness. In terms of boundedness of the height, we get an optimal statement for a curve $X$ with $d=2$. We also deal with the fattenings $\varGamma_\varepsilon$ introduced by Poonen. The proof rests on a suitably uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $A$.

Wir entwickeln eine Strategie zum Beweis der Positivität der Koeffizienten von Kazhdan-Lusztig-Polynomen für beliebige Coxeter-Gruppen.

We develop a strategy to prove the positivity of coefficients of Kazhdan–Lusztig polynomials for arbitrary Coxeter groups.

We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a non-commutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz–Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various authors. We prove that bivariant equivariant periodic cyclic homology is homotopy invariant, stable and satisfies excision in both variables. Moreover, we construct the exterior product which generalizes the obvious composition product. Finally, we prove a Green–Julg theorem in cyclic homology for compact groups and the dual result for discrete groups.

We introduce and study a metric notion for trees and develop a fine structure theory for the corresponding class of Lipschitz trees. We also relate this structure theory to a conjecture of Shelah about the existence of a finite basis for a class of linear orderings and solve an old problem of Laver about well-quasi-ordering a certain class of trees.