Formula omitted]-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series

Abstract

Let G0 be a simply connected non-compact real simple Lie group and let K0 be a maximal compact subgroup of G0. Suppose that K0 is semisimple and that rank(K0)=rank(G0). Let Δ+ be a Borel–de Siebenthal positive root system and let πλ be the Borel–de Siebenthal discrete series of G0 with Harish-Chandra parameter λ. One has a certain subgroup L0⊂K0 so that K0/L0 is an irreducible Hermitian symmetric space. Also, there is a holomorphic discrete series πλ′ of K0⁎, the non-compact dual of K0, with Harish-Chandra parameter λ′:=λ−(1/2)∑α, where the sum is over non-compact roots in Δ+. We prove that there are infinitely many L0-types common to πλ and πλ′ under certain hypotheses.