Szegő maps and highest weight representations

Abstract

Let G be a connected noncompact simple Lie group with finite center and let A 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 (7-equivariant Szegό-type integral operator Sτ such that Sτ maps the K-finite vectors in W(τ) onto an irreducible highest weight $module L(τ). Of primary concern here are those representations τ which are reduction points. For such τ, we construct certain systems 3lτ of ΰ-equivariant differential operators and then utilize 3fτ to establish the infinitesimal irreducibility of the image of Sτ. 1. Introduction. 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. The main purpose of this article is to realize each irreducible highest weight representation of G as the image of a G-equivariant quotient map defined on principal series representations. To make this more precise, recall that each irreducible highest weight representation πτ of G is parametrized by an irreducible unitary representation τ of K. Let C°°(G, τ) denote the space of τ-covariant C°°-functions on G. We associate to τ a principal series representation W(τ) and a Szego map Sτ: W(τ) -> C°°(G, τ) having the property that the ^-finite vectors in W(τ) are mapped onto an irreducible g-module equivalent to the derived action of πτ. In the case of discrete series and limits thereof, this type of result was proved by Knapp and Wallach [16] in the general setting where G is a semisimple equirank Lie group with finite center. The main result here is that the irreducibility of the image of *S τ persists for all highest weight representations. Moreover, for certain τ called reduction points, the irreducibility of Image (Sτ) is proved by showing this space is annihilated by a system Qίτ of Gequivariant differential operators. The system 3τ somewhat parallels the role of the Schmid operator in the Knapp and Wallach result. The realization of distinguished representations as irreducible images of quotient maps is a recurring theme in the literature which