The explicit theta correspondence for reductive dual pairs $(Sp(p,q),O^{*}(4))$

TL;DR

This paper explicitly determines the theta correspondence for all irreducible modules of $O^{*}(4)$ to $Sp(p,q)$ using Langlands parameters, infinitesimal characters, and lowest $K$-types.

Contribution

It provides a complete explicit description of the theta lifts for $O^{*}(4)$ to $Sp(p,q)$ for all non-negative integers p and q.

Findings

Explicit formulas for theta lifts of all irreducible $O^{*}(4)$ modules.

Determination of theta lifts via infinitesimal characters and lowest $K$-types.

Applicable to all non-negative p and q.

Abstract

For every irreducible Harish-Chandra module of $O^{*}(4)$, we determine its theta lift to $Sp(p,q)$ in terms of the Langlands parameter, for all non-negative integers $p$ and $q$. Our strategy is to determine the desired theta lifts by their infinitesimal characters and lowest $K$-types.