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.

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.