Let $G$ be a complex, semisimple, simply connected algebraic group with Lie algebra ${\frak g}$. We extend scalars to the power series field in one variable ${\rm C}((\pi))$, and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of ${\frak g} \otimes {\rm C}((\pi))$, i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig's topological construction of Springer's representations to the affine context.

Crystal base of the level 0 part of the modified quantum affine algebra $\tilde{U}_q (\widehat{{\frak s}{\frak l}_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.

Abstract. Explicit models are constructed for irreducible *-representations of the quantised universal enveloping algebra $U_q({\frak g}{\frak l}(n))$. The irreducible decomposition of these modules with respect to the subalgebra $U_q({\frak g}{\frak l}(n-1))$ is given, and the corresponding spherical and associated spherical elements are determined in terms of little $q$-Jacobi polynomials. This leads to a proof of an addition theorem for the spherical elements, the so-called $q$-disk polynomials.

The main goal of this Letter is to give a new proof of a result conjectured by D. De George and N. Wallach and proved by P. Delorme and to extend it to the $S$-arithmetic setting. The proof is based on a density principle.

We study a family of K3 surfaces which have a big automorphism group. We begin with generalisations of Silverman's results: construction of canonical heights, density of rational points in one orbit,… We continue the study in estimating the density of rational points on the orbiting rational curves; this estimate is compatible with Batyrev–Manin conjecture. Moreover we settle, under more geometric hypothesis, the number of rational points of such surfaces of bounded height.

We study residues on a complete toric variety $X$, which are defined in terms of the homogeneous coordinate ring of $X$. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When $X$ is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent $X$ as a quotient $(Y\setminus\{0\})/C\ast$ such that the toric residue becomes the local residue at 0 in $Y$.

We present some basic identities for hypergeometric functions associated with the integrals of Euler type. We give a geometrical proof for formulae such as the identity between the single and double integrals expressing Appell's hypergeometric series $F_1 (a, b, b^{\prime} c; x, y)$.

We proove an analogue of the Beilinson-Bernstein Theorem, in the case of the projective space, for the ring of arithmetical differential operators introduced by P. Berthelot.

In this paper, we study representations of a Galois group of a local field absolutely unramified whose residue field is perfect of characteristic $p$ and give two ways of recognize crystalline representations with weights between $r$ and $r+p-1$. The first one is the description of terms of weakly admissible filtered modules proved by J.-M. Fontaine and G. Laffaille (Ann Sc ENS 1982): here a new proof of this result is proposed (second chapter). The second criterion (third chapter) is the equivalence between crystalline representations with weights between $r$ and $r+p-1$ and representations of finite ‘cr-height’ ${\leqslant} p-1$, result announced by J.-M. Fontaine in a paper edited a few years ago where he introduced the category of $(\varphi, \Gamma)$-modules.

This paper generalizes the definition of mixed perverse sheaves to schemes of finite type over a number field. Their basic properties, e.g., characterization of simple objects, are shown.