On the Gelfand property for complex symmetric pairs

Abstract

We first prove, for pairs consisting of a simply connected complex reductive group together with a connected subgroup, the equivalence between two different notions of Gelfand pairs. This partially answers a question posed by Gross, and allows us to use a criterion due to Aizenbud and Gourevitch, and based on Gelfand-Kazhdan's theorem, to study the Gelfand property for complex symmetric pairs. This criterion relies on the regularity of the pair and its descendants. We introduce the concept of a pleasant pair, as a means to prove regularity, and study, by recalling the classification theorem, the pleasantness of all complex symmetric pairs. On the other hand, we prove a method to compute all the descendants of a complex symmetric pair by using the extended Satake diagram, which we apply to all pairs. Finally, as an application, we prove that eight out of the twelve exceptional complex symmetric pairs, together with the infinite family $(\textrm{Spin}_{4q+2}, \textrm{Spin}_{4q+1})$, satisfy the Gelfand property, and state, in terms of the regularity of certain symmetric pairs, a sufficient condition for a conjecture by van Dijk and a reduction of a conjecture by Aizenbud and Gourevitch.