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

By Avraham Aizenbud, D. Gourevitch, Eitan Sayag. Compositio Mathematica, 2007

50 citations recorded by the source corpus.

Abstract Let F be an arbitrary local field. Consider the standard embedding $\mathrm {GL}_n(F) \hookrightarrow \mathrm {GL}_{n+1}(F)$ and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), $\dim Hom_{\mathrm {GL}_n(F)}(E,\mathbb {C}) \leqslant 1$. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e63f2c5f9878daa4eeaa2db4aeb6cd4f9ba8a462