K-invariant cusp forms for reductive symmetric spaces of split rank one

Let {G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for {G/H}. We show that the space of cusp forms coincides with the closure of the space of K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of {G/H}.

Integral Representation for U3 × GL2

Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin–Selberg type for groups G×GLn, where G is of split rank n. Here we show that their method can equally well be applied to the product U3 × GL2, where U3 denotes the quasisplit unitary group in three variables. As an application, we describe which cuspidal automorphic representations of U3 occur in the Siegel induced residual spectrum of the quasisplit U4.

Theta Lifts of Tempered Representations for Dual Pairs (Sp2n,O(V))

This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs (Sp2n,O(V)) over a non-archimedean field F of characteristic different than 2, where n is the split rank of Sp2n and the dimension of the space V (over F) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.