-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series
Abstract Let G 0 be a simply connected non-compact real simple Lie group and let K 0 be a maximal compact subgroup of G 0 . Suppose that K 0 is semisimple and that rank ( K 0 ) = rank ( G 0 ) . Let Δ + be a Borel–de Siebenthal positive root system and let π λ be the Borel–de Siebenthal discrete series of G 0 with Harish-Chandra parameter λ. One has a certain subgroup L 0 ⊂ K 0 so that K 0 / L 0 is an irreducible Hermitian symmetric space. Also, there is a holomorphic discrete series π λ ′ of K 0 ⁎ , the non-compact dual of K 0 , 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 L 0 -types common to π λ and π λ ′ under certain hypotheses.