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

Abstract

For every irreducible Harish-Chandra module of $O^{\ast }(4)$, 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.