On the Gelfand property for complex symmetric pairs

TL;DR

This paper investigates the Gelfand property for complex symmetric pairs, establishing criteria for regularity, classifying pleasant pairs, and verifying the property for most exceptional cases, advancing understanding of representation theory.

Contribution

It introduces the concept of pleasant pairs, develops methods to compute descendants using Satake diagrams, and proves the Gelfand property for most complex symmetric pairs, partially answering open questions.

Findings

Eight of twelve exceptional pairs satisfy the Gelfand property.

A criterion based on regularity and descendants determines the Gelfand property.

Classification of pleasant pairs aids in proving regularity 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.