Some regular symmetric pairs

Abstract

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 C.In this paper we show that the pairs (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 (C), O n (C)) and (O n+m (C), O n (C) × O m (C)) are Gelfand pairs.

A partial analog of the integrability theorem for distributions on p-adic spaces and applications

“…Defects of symmetric pairs. In this subsection we review some tools developed in [AG2] and [AG3] that enable to prove that a symmetric pair is special.…”

“…In subsubsection 5.1.2 we review a technique from [AG2] for proving that a given pair is a Gelfand pair. In subsubsections 5.1.3-5.1.7 we review a technique from [AG2] and [AG3] for proving that a given symmetric pair is a Gelfand pair.…”

Gelfand pairs(Sp4n(F),Sp2n(Zhang1 2010Journal of Number Theory7000View full textAdd to dashboardCite

“…To calculate the descendants of symmetric pairs, we refer calculation for some symmetric pairs in [AG2], in which they apply the method of computing centralizers of semi-simple elements of classical groups in [SS]. In my previous proof in [Z], I wrote down explicit representatives of all the semi-simple H -conjugate classes in G σ and calculate their centralizers in G and H respectively.…”

“…In my previous proof in [Z], I wrote down explicit representatives of all the semi-simple H -conjugate classes in G σ and calculate their centralizers in G and H respectively. After Aizenbud and Gourevitch's easier calculation in [AG2], I adopt their argument to calculate the descendants of our symmetric pairs. Let (G, H, θ) be one of following symmetric pairs,…”

On the Gelfand property for complex symmetric pairs

“…As an application, this criterion, together with a case-by-case proof of the equivalence between Gelfand pairs and Gelfand pairs à la Gross, was used to prove that, for any local field F , the pairs (GL n+k (F ), GL n (F ) × GL k (F )) and (GL n (E), GL n (F )), for E a quadratic extension of F , are Gelfand pairs. Furthermore, the same was done in [AG10] to prove that the complex pairs (GL n , O n ) and (O n+m , O n × O m ) are Gelfand pairs (it had already been proved that (SO n , SO n−1 ) is a unitary Gelfand pair [AvD06] and a Gelfand pair [AGS09]). It was conjectured by Aizenbud and Gourevitch [AG09,Conj.…”