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

2008; Springer Science+Business Media; Volume: 261; Issue: 2 Linguagem: Inglês

10.1007/s00209-008-0318-5

1432-1823

Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag,

Finite Group Theory Research

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).