Vector-valued Gelfand-Kazhdan criterion

TL;DR

This work generalizes the Gelfand–Kazhdan criterion to vector-valued settings within the relative Langlands framework, enabling multiplicity-one results beyond one-dimensional $\omega$. It introduces a vector-valued GK criterion that bounds $\dim \mathrm B_{H_1}(\pi\times \omega_1, \mathbb{C}) \cdot \dim \mathrm B_{H_2}(\pi^{\vee}\times \omega_2, \mathbb{C})$ by $1$ under a symmetry condition expressed via a topological anti-automorphism and isomorphisms between $\omega_1$ and $\omega_2$, using Schwartz densities $\mathrm D^{\varsigma}(G)$. The criterion is then applied to establish multiplicity-one for local Asai Rankin–Selberg periods by treating $(\mathrm{GL}_n(\mathrm k'), \mathrm D^{\varsigma}(\mathrm k^{n\times 1}))$ as a Gelfand pair, and it yields a framework for disjointness of local periods. Overall, the results provide a robust tool for analyzing multiplicity-one phenomena in the relative Langlands program and their connections to $L$-functions.

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.

Theorems & Definitions (16)

• Theorem 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 1.6
• Remark 1.7
• Lemma 2.1
• Lemma 2.2
• proof