(O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F

TL;DR

This paper proves that the pair (O(V), O(W)) forms a Gelfand pair over any local field by showing invariance of distributions under transposition, extending previous results with a new proof approach.

Contribution

It provides a new proof that (O(V), O(W)) is a Gelfand pair over any local field, generalizing earlier results and confirming invariance properties of distributions.

Findings

Invariant distributions are symmetric under transposition.

The pair (O(V), O(W)) is a Gelfand pair over all local fields.

Implication for multiplicity-one property in representations.

Abstract

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $W=V \oplus Fe$ with the form Q extending q with Q(e)=1. Consider the standard embedding of O(V) into O(W) and the two-sided action of $O(V)\times O(V)$ on $O(W)$. In this note we show that any $O(V)\times O(V)$-invariant distribution on O(W) is invariant with respect to transposition. This result was earlier proven in a bit different form in [vD] for F=R, in [AvD] for F=C and in [BvD] for p-adic fields. Here we give a different proof. Using results from [AGS], we show that this result on invariant distributions implies that the pair (O(V),O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Frechet representation E of O(W) we have $dim Hom_{O(V)}(E,C) \leq 1.$ A stronger result for p-adic fields is obtained in [AGRS].