Degenerate principal series representations of Sp(p, q)

Abstract

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL q (ℍ)×Sp(p−q). Forseℂ andμ a highest weight of Sp(p−q), let пs,µ be the representation ofP such that its restriction toN is trivial and $$\pi s,\mu |L = \det _q^s $$ ⊠T p-q μ , where det q is the determinant character of GL q (ℍ) andT p-q μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI p,q(s, μ) be the Harish-Chandra module of the induced representation Ind G $$\begin{array}{*{20}c} G \\ P \\\end{array}\pi _{s,\mu } $$ . In this paper, we shall determine the module structure and unitarity ofI p, q(s, μ).