The nonarchimedean theta correspondence for GSp(2) and GO(4)

In this paper we consider the theta correspondence between the sets Irr(GSp(2, k)) and Irr(GO(X)) when k is a nonarchimedean local field and dim(k) X = 4. Our main theorem determines all the elements of Irr(GO(X)) that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of Irr(O(X)) that occur in the theta correspondence between Irr(Sp(2, k)) and Irr(O(X)). We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.