Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(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), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.
- Mathematics - Representation Theory