Degenerate principal series representations of U(p, q) and spin 0(p, q)

Abstract

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 (equation presented). Let χ be a one-dimensional character of M and τμ 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 multiplication sign in box τμ. Let Ip,q be the Harish-Chandra module of the induced representation IndP Gπ χ,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.