Derivatives and L-Functions for GLn

Abstract

Bernstein and Zelevinsky analyzed the representations of GLn over a non-archimedean local field by restriction to the mirabolic subgroup Pn. In doing this, they defined the notion of the derivatives of a representation of GLn. For generic representations, we relate these derivatives to the restriction of Whittaker functions to the embedded GLk that appear in the theory of local L-functions for GLn × GLm. Combining this realization of derivatives with a deformation argument, we are able to derive the (known) formulas for the local L-functions for GLn × GLm in terms of their exceptional poles, which depend on invariant pairings between the derivatives of the representations of GLn and GLm. We expect this way of computing L-functions will be applicable to other L-functions for GLn, such as the exterior square or symmetric square, as well as the local archimedean situation.