Uniqueness and disjointness of Klyachko models

We show the uniqueness and disjointness of Klyachko models for GL(n,F) over a non-archimedean local field F. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum

(GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfand pair. Namely, for any irreducible admissible representation $(\pi,E)$ of (GL(n+1,F), dimHom_{GL(n,F)}(E,\cc) \leq 1. For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds.Comment: v3: Archimedean Localization principle excluded due to a gap in its proof. Another version of Localization principle can be found in arXiv:0803.3395v2 [RT]. v4: an inaccuracy with Bruhat filtration fixed. See Theorem 4.2.1 and Appendix

The harmonic analysis of lattice counting on real spherical spaces

By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak.Comment: Extended and revised version with 4 additional pages. To appear in Documenta mat