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

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 $\mathrm{O}(V) \hookrightarrow \mathrm{O}(W)$ and the two-sided action of $\mathrm{O}(V)\times \mathrm{O}(V)$ on $\mathrm{O}(W)$ . In this note we show that any $\mathrm{O}(V) \times \mathrm{O}(V)$ -invariant distribution on $\mathrm{O}(W)$ is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for $F={\mathbb{R}}$ , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for $F={\mathbb{C}}$ and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), 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 $\mathrm{O}(W)$ we have $dim Hom_{\mathrm{O}(V)}(E,\mathbb{C}) \leq 1.$ A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).