Towards a Goldberg-Shahidi pairing for classical groups

Let G be either an orthogonal, a symplectic or a unitary group over a local field F and let P = MN be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G 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 less than or equal to 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. 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.