On the local theta representation

TL;DR

This paper develops an algebraic framework for theta correspondences between representations of groups and explores their extension to larger groups, with applications to reductive dual pairs of similitude groups in non-archimedean settings.

Contribution

It introduces a new algebraic approach to theta correspondence and investigates conditions for extending these correspondences to larger groups.

Findings

Established criteria for extending theta correspondences to bigger groups.

Applied the framework to reductive dual pairs of similitude groups in non-archimedean fields.

Provided insights into the structure of theta representations in this context.

Abstract

We study the algebraic framework in which one can define, in the manner of the theta correspondence, a correspondence between representations of two locally profinite groups $H_1$, $H_2$. In particular, we examine when and how such a correspondence can be extended to bigger groups $G_1$, $G_2$ containing $H_1$, $H_2$ respectively as normal subgroups. As an application, we discuss the theta correspondence for a reductive dual pair of the similitude groups in the non-archimedean case.