OF U ( p , q ) AND Spin 0 ( p , q )

Let p > q and let G be the group U(p, q) or Spin0(p, q). Let P = LN be the maximal parabolic subgroup of G with Levi subgroup L ∼=M × U where (M,U) = { (GLq(C),U(p− q)) if G = U(p, q) (GLq (R),Spin(p− q)) if G = Spin0(p, q). Let χ be a 1 dimensional character of M and τ an irreducible representation of U with highest weight μ. Let πχ,μ be the representation of P which is trivial on N and πχ,μ|L = χ τ. Let Ip,q be the Harish-Chandra module of the induced representation IndPπχ,μ. In this paper, we shall determine (i) the reducibility of Ip,q, (ii) the K-types of all the irreducible subquotients of Ip,q when it is reducible, where K is the maximal compact subgroup of G, (iii) the module diagram of Ip,q (from which one can read off the composition structure), and (iv) the unitarity of Ip,q and its subquotients. Except in the cases q = p− 1 and q = 1, Ip,q is not K-multiplicity free.