The explicit theta correspondence for reductive dual pairs (Sp(p,q),O⁎(4))

For every irreducible Harish–Chandra module of O⁎(4)O⁎(4), we determine its theta lifts to Sp(p,q)Sp(p,q) in terms of the Langlands parameters, 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.