Boundary value maps, Szegö maps and intertwining operators

Abstract:

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