Restriction of Discrete Series of SU(2,1) to S(U(1) × U(1,1))

The group G = SU(2, 1) possesses nonempty holomorphic, antiholomorphic, and nonholomorphic discrete series. The restriction of these discrete series to the spherical subgroup G1 = S(U(1) × U(1, 1)) is studied in this paper. We prove that direct integral decomposition of any restricted discrete series of G is multiplicity free. In (J. Funct. Anal.103 (1992), 352-371), J. Vargas claimed that there was no discrete part in the direct integral of any restricted nonholomorphic discrete series. Unfortunately, his proof was wrong. Our argument in Section 5 shows that there are infinitely many discrete series of G1 occurring in the discrete part for any restricted nonholomorphic discrete series of G, and both the discrete and the continuous parts are not empty. The decomposition in Section 5 confirms a conjecture of B. Gross (B. H. Gross and D. Prasad, Canad. J. Math.44, No. 5 (1992), 974-1002) for these groups. Our main interest is of course in the restriction of nonholomorphic discrete series, but for completeness, we consider all discrete series of G.