Boundary value maps, Szegö maps and intertwining operators

Abstract

We consider one series of unitarizable representations, the cohomological induced modules A q ( λ ) A_{\mathfrak {q}}(\lambda ) with dominant regular infinitesimal character. The minimal K K -type ( τ , V ) (\tau , V) of A q ( λ ) A_{\mathfrak {q}}(\lambda ) determines a homogeneous vector bundle V τ ⟶ G / K V_{\tau } \longrightarrow G/K . The derived functor modules can be realized on the solution space of a first order differential operator D l λ \mathcal {D}_{\mathfrak {l}}^{\lambda } on V τ V_{\tau } . Barchini, Knapp and Zierau gave an explicit integral map S \mathcal {S} from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle V τ ⟶ G / K V_{\tau } \longrightarrow G/K . In this paper we study the asymptotic behavior of elements in the image of S \mathcal {S} . We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map S \mathcal {S} and a passage to boundary values.