A Gelfand Pair for Any Quadratic Space V over a Local Field

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let W = V ⊕ Fe with the form Q extending q with Q(e) = 1. Consider the standard embedding O(V ) ↪→ O(W ) and the two-sided action of O(V )×O(V ) on O(W ). In this note we show that any O(V )×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 Fréchet representation (π,E) of O(W ) we have dimHomO(V )(E,C) ≤ 1. A stronger result for p-adic fields is obtained in [AGRS07].