Degenerate Principal Series Representations of U( p , q ) and Spin 0 ( p , q )

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. 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 $L\cong M\times U$ where$\openup2(M,U)=\left \{ \matrix{ ({\rm GL} _q({\mathbb C}),{\rm U}(\,p-q)),\hfill \mbox { if } G={\rm U}(\,p,q), \hfill\cr ({\rm GL} ^+_q({\mathbb R}),{\rm Spin}(\,p-q)), \mbox { if } G={\rm Spin}_0(\,p,q). }\right.$Let χ be a one-dimensional character of M and τμ an irreducible representation of U with highest weight μ. Let $\pi_{\chi,\mu}$ be the representation of P which is trivial on N and $\pi_{\chi,\mu}|_L=\chi\boxtimes \tau ^\mu$. Let $I_{p,q}$ be the Harish-Chandra module of the induced representation ${\rm Ind}_{P}^{G} \pi_{\chi,\mu}$. In this paper, we shall determine (i) the reducibility of $I_{p,q}$, (ii) the K-types of all the irreducible subquotients of $I_{p,q}$ when it is reducible, where K is the maximal compact subgroup of G, (iii) the module diagram of $I_{p,q}$ (from which one can read off the composition structure), and (iv) the unitarity of $I_{p,q}$ and its subquotients. Except in the cases $q=p-1$ and $q=1$, $I_{p,q}$ is not K-multiplicity free.