Some regular symmetric pairs

Aizenbud, Avraham; Gourevitch, Dmitry

doi:10.1090/S0002-9947-10-05008-7

Some regular symmetric pairs
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Trans. Amer. Math. Soc. 362 (2010), 3757-3777 Request permission

Abstract:

In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over $\mathbb {C}$.

In this paper we show that the pairs $(GL(V),O(V)), (GL(V),U(V))$, $(U(V),O(V)), (O(V \oplus W),O(V) \times O(W)), (U(V \oplus W),U(V) \times U(W))$ are regular, where $V$ and $W$ are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs $(GL_n(\mathbb {C}),O_n(\mathbb {C}))$ and $(O_{n+m}(\mathbb {C}),O_n(\mathbb {C}) \times O_m(\mathbb {C}))$ are Gelfand pairs.

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