
Journal of Lie Theory 24 (2014), No. 2, 475501 Copyright Heldermann Verlag 2014 Borelde Siebenthal Discrete Series and Associated Holomorphic Discrete Series Pampa Paul Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600 113, India pampa@imsc.res.in Komaranapuram N. Raghavan Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600 113, India knr@imsc.res.in Parameswaran Sankaran Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600 113, India sankaran@imsc.res.in [Abstractpdf] Let $G_0$ be a simply connected noncompact real simple Lie group with maximal compact subgroup $K_0$. Assume that rank$(G_0)$ = rank$(K_0)$ so that $G_0$ has discrete series representations. If $G_0/K_0$ is Hermitian symmetric, one has a relatively simple discrete series of $G_0$, namely the holomorphic discrete series of $G_0$. Now assume that $G_0/K_0$ is not a Hermitian symmetric space. In this case, one has the class of Borelde Siebenthal discrete series of $G_0$ defined in a manner analogous to the holomorphic discrete series. We consider a certain circle subgroup of $K_0$ whose centralizer $L_0$ is such that $K_0/L_0$ is an irreducible compact Hermitian symmetric space. Let $K_0^*$ be the dual of $K_0$ with respect to $L_0$. Then $K_0^*/L_0$ is an irreducible noncompact Hermitian symmetric space dual to $K_0/L_0$. In this article, to each Borelde Siebenthal discrete series of $G_0$, we will associate a holomorphic discrete series of $K_0^*$. Then we show the occurrence of infinitely many common $L_0$types between these two discrete series under certain conditions. Keywords: Discrete series, admissibility, relative invariants, branching rule, LSpaths. MSC: 22E46, 17B10 [ Fulltextpdf (487 KB)] for subscribers only. 