Towards a Goldberg-Shahidi pairing for classical groups

Abstract

Let G 1 be an orthogonal, symplectic or unitary group over a local field and let P = M N be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G 1 and a general linear group, acting on vector spaces X and W, respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M, assuming dim W ≤ dim X , excluding only the symplectic case with dim W odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G 1. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.