The SL(1,D)-distinction problem

TL;DR

This paper investigates the distinction problem for a specific symmetric pair involving SL(2,E) and SL(1,D) over local fields, using local theta correspondences between quaternionic groups.

Contribution

It introduces a novel approach to the distinction problem by applying local theta correspondences between quaternionic Hermitian and skew-Hermitian groups.

Findings

Characterizes when the representation is distinguished using theta correspondence

Provides new criteria for distinction in the context of quaternionic groups

Enhances understanding of the structure of representations over local fields

Abstract

We use the local theta correspondences between the quaternionic Hermitian groups and the quaternionic skew-Hermitian groups to understand the distinction problem for the symmetric pair SL(2,E)/SL(1,D), where E is a quadratic field extension of a nonarchimedean local field extension F and D is a 4-dimensional division quaternion algebra over F