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

Abstract

The group G = SU(2, 1) possesses nonempty holomorphic, antiholomorphic, and nonholomorphic discrete series. The restriction of these discrete series to the spherical subgroup G 1 = 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 G 1 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.