The space of degenerate Whittaker models for general linear groups over a finite field

Abstract

Let G=GL_2n(F), where F is a finite field, and P the (n,n) parabolic in G with Levi subgroup GL_n(F)×GL_n(F) and unipotent radical N=M_n(F). Let ψ₀ be a nontrivial additive character ψ₀: F→C^∗. Let ψ(X)=ψ₀(tr X) be the additive character on N=M_n(F). Let π be an irreducible admissible representation of G. Let π_N,ψ, be the largest subspace of π on which N operates via ψ. Since tr(gXg^-1=tr(X), it follows that π_N,ψ is a representation space for H=ΔGL_n(F)→GL_n(F)×GL_n(F). The space π_N,ψ, is referred to as the space of degenerate Whittaker models, or sometimes also as the twisted Jacquet functor of the representation π. The aim of this work is to calculate this for cuspidal representations of GL_2n(F).