A general form of Gelfand-Kazhdan criterion

TL;DR

This paper generalizes the Gelfand-Kazhdan criterion by formalizing matrix coefficients for distributional vectors in real reductive group representations, broadening its applicability.

Contribution

It introduces a formal framework for matrix coefficients of distributional vectors and proves a generalized Gelfand-Kazhdan criterion for real reductive groups.

Findings

Established a formal notion of matrix coefficients for distributional vectors.

Proved a generalized Gelfand-Kazhdan criterion in broad settings.

Enhanced understanding of representation theory for real reductive groups.

Abstract

We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group, which consist of generalized functions on the group. As an application, we state and prove a Gelfand-Kazhdan criterion for a real reductive group in very general settings.