The Gelfand-Kazhdan criterion as a necessary and sufficient criterion

TL;DR

This paper demonstrates that the Gelfand-Kazhdan criterion is both necessary and sufficient for the Gelfand property under certain conditions, extending the understanding of this criterion across various group types.

Contribution

It establishes the Gelfand-Kazhdan criterion as a necessary and sufficient condition for the Gelfand property, broadening its applicability to finite, p-adic, and real groups.

Findings

Gelfand-Kazhdan criterion is necessary for the Gelfand property.

The criterion's sufficiency is proven under certain conditions.

Partial extension of results to p-adic and real groups.

Abstract

We show that under certain conditions the Gelfand-Kazhdan criterion for the Gelfand property is a necessary condition. We work in the generality of finite groups, however part of the argument carries over to p-adic and real groups.