Prasad, Dipendra (2000) The space of degenerate Whittaker models for general linear groups over a finite field International Mathematics Research Notices, 2000 (11). pp. 579595. ISSN 10737928

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Official URL: http://imrn.oxfordjournals.org/content/2000/11/579...
Related URL: http://dx.doi.org/10.1155/S1073792800000313
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 ψ_{0} be a nontrivial additive character ψ_{0}: F→C^{∗}. Let ψ(X)=ψ_{0}(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).
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