Let G be a connected
noncompact simple Lie group with finite center and let K be a maximal compact
subgroup of G. Assume the space G∕K is Hermitian symmetric. We associate to each
irreducible representation τ of K a principal series representation W(τ) and a
G-equivariant Szegö-type integral operator Sτ such that Sτ maps the K-finite
vectors in W(τ) onto an irreducible highest weight g-module L(τ). Of primary
concern here are those representations τ which are reduction points. For such τ, we
construct certain systems 𝒟τ of G-equivariant differential operators and
then utilize 𝒟τ to establish the infinitesimal irreducibility of the image of
Sτ.
