Algebraic aspects of branching laws for holomorphic discrete series representations (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

Abstract

We consider the branching problem of holomorphic discrete series representations and their analytic continuation with respect to a symmetric subgroup of anti-holomorphic type. The main purpose is to prove the irreducibility of a (mathrm{g}oplus mathfrak{g}', $Delta$(G'))-module Homc(V, V')_{$Delta$(G')} for the underlying Harish-Chandra module V of a holomorphic discrete series representation and a generic (mathrm{g}', K')-module V'. Comparing the branching law of unitary representations to this result, we conjecture that the irreducibility of the (mathfrak{g}oplus mathfrak{g}', $Delta$(G'))-module Homc(V, V')_{triangle(G')} is related to the existence of a discrete spectrum.