In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an admissible irreducible representation of GL(n + 1), when restricted to GL(n) decomposes with multiplicity one. Similar Theorems are obtained for orthogonal or unitary groups.

Let G be a semisimple algebraic group defined over Q p , and let Γ be a compact open subgroup of G(Q p ). We relate the asymptotic representation theory of Γ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both:1. We prove that there is a constant C, independent of G, such that the number of n-dimensional representations of Γ grows slower than n C , confirming a conjecture of Larsen and Lubotzky. In fact, we can take C = 3 · dim(E 8 ) + 1 = 745. We also prove the same bounds for groups over local fields of large enough characteristic.2. We prove that the coarse moduli space of G-local systems on a smooth projective curve of genus at least C/2 + 1 = 374 has rational singularities.For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities.

Abstract. In the first part of the paper we generalize a descent technique due to Harish-Chandra to

In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on R n and build some additional tools which are important in representation theory.

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set.We prove this result for the Archimedean and the non-Archimedean case in a uniform way. The Archimedean case was proved in [Ber1]. The non-Archimedean case was proved in [HK] and [CL1, CL2]. Our method is different from those described in [Ber1, HK, CL1, CL2]. It is based on Hironaka's desingularization theorem, unlike [Ber1] which is based on the theory of D-modules and [HK,CL1,CL2] which is based on model theory.Our method also gives bounds on the open dense set where the Fourier transform is smooth and moreover, on the wave front set of the Fourier transform. These bounds are explicit in terms of resolution of singularities and field-independent.We also prove the same results on the Fourier transform of more general measures of algebraic origins.