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 dimk 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. Suppose k is a nonarchimedean local field of characteristic zero and odd residual characteristic, X is an even dimensional nondegenerate symmetric bilinear space over k and n is a nonnegative integer. Let ω be the Weil representation of Sp(n, k)×O(X) corresponding to a fixed choice of nontrivial additive character of k, and let RX(Sp(n, k)) be the set of elements of Irr(Sp(n, k)) that are nonzero quotients of ω; similarly define Rn(O(X)). By [W], the condition that π ⊗C σ be a nonzero quotient of ω for π in RX(Sp(n, k)) and σ in Rn(O(X)) defines a bijection between RX(Sp(n, k)) and Rn(O(X)). By [R], the extension of ω to the subgroup R of GSp(n, k) × GO(X) consisting of pairs whose entries have the same similitude factor also defines a well behaved correspondence between Irr(GSp(n, k)) and Irr(GO(X)). Here, GSp(n, k) is the subgroup of elements of GSp(n, k) whose similitude factors lie in the group of similitude factors of the elements of GO(X); thus, GSp(n, k) is of index at most two in GSp(n, k) and contains Sp(n, k). To be more precise about the correspondence, let RX(GSp(n, k)) be the set of elements of Irr(GSp(n, k)) whose restrictions to Sp(n, k) are multiplicity free and have a constituent in RX(Sp(n, k)); similarly define Rn(GO(X)). Then by [R] the condition HomR(ω, π ⊗C σ) 6= 0 defines a bijection between RX(GSp(n, k)) and Rn(GO(X)). Granted the theta dichotomy conjecture, if dimkX ≤ 2n, then GSp(n, k) may be replaced by GSp(n, k) in this result. Received by the editors June 3, 1996 and, in revised form, February 6, 1997. 1991 Mathematics Subject Classification. Primary 11F27; Secondary 22E50.