Theta correspondence for symplectic-orthogonal and metaplectic-orthogonal p-adic dual pairs

TL;DR

This paper fully characterizes the Howe theta correspondence for symplectic-orthogonal and metaplectic-orthogonal dual pairs over nonarchimedean local fields, detailing first occurrence indices and Langlands parameter descriptions.

Contribution

It provides a complete description of the theta correspondence for these dual pairs, including explicit first occurrence indices and Langlands parameter characterizations.

Findings

Determined first occurrence indices for all irreducible admissible representations.

Described small theta lifts explicitly in terms of Langlands parameters.

Established a comprehensive framework for the Howe correspondence in this setting.

Abstract

In this paper, we completely describe the Howe correspondence for the dual pairs from the title over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.