Vector-valued Gelfand-Kazhdan criterion

TL;DR

This paper generalizes the Gelfand-Kazhdan criterion to a vector-valued form within the relative Langlands program, enabling new proofs of multiplicity-one properties for local periods like the Asai Rankin-Selberg periods.

Contribution

It introduces a vector-valued Gelfand-Kazhdan criterion applicable to the relative Langlands program, broadening the scope of multiplicity-one results.

Findings

Established multiplicity-one for local Asai Rankin-Selberg periods

Generalized Gelfand-Kazhdan criterion to vector-valued setting

Enhanced tools for the relative Langlands program

Abstract

The Gelfand-Kazhdan criterion is a fundamental tool for studying multiplicity-one properties of local periods of representations. However, it does not apply to many cases arising in the relative Langlands program. Generalizing the usual Gelfand-Kazhdan criterion, we formulate and prove a vector-valued Gelfand-Kazhdan criterion that fits into the general framework of the relative Langlands program. As an illustration of its effectiveness, we establish the multiplicity-one property for the local Asai Rankin-Selberg periods.