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Borel-de Siebenthal discrete series and associated holomorphic discrete series

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, there exists a relatively simple discrete series of G_0, called holomorphic discrete series. Now assume that G_0/K_0 is not Hermitian symmetric. In this case, we can define Borel-de Siebenthal discrete series of G_0 analogous to 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 non-compact Hermitian symmetric space dual to K_0/L_0. To each Borel-de Siebenthal discrete series of G_0, we can associate a holomorphic discrete series of (K_0)*. In this article, we address occurrence of common L_0-types between these two discrete series under certain conditions.