On the singularities of residual intertwining operators

Abstract.

Let G be a reductive algebraic group defined over ${\Bbb Q}$. Let P, P' be parabolic subgroups of G, defined over ${\Bbb Q}$, and let $t \in W({\frak a}_{P}, {\frak a}_{P'})$. In this paper we study the intertwining operator $M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}}$, acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of $M_{P' \vert P}(t,\lambda)$ is a meromorphic function of order $\le n + 1$, where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.