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Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 609–633 | Cite as

On certain degenerate Whittaker Models for cuspidal representations of \({\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}\)

  • Ofir Gorodetsky
  • Zahi HazanEmail author
Article
  • 98 Downloads

Abstract

Let \(\pi \) be an irreducible cuspidal representation of \(\mathrm {GL}_{kn}(\mathbb {F}_q)\). Assume that \(\pi = \pi _{\theta }\), corresponds to a regular character \(\theta \) of \(\mathbb {F}_{q^{kn}}^{*}\). We consider the twisted Jacquet module of \(\pi \) with respect to a non-degenerate character of the unipotent radical corresponding to the partition \((n,n,\ldots ,n)\) of kn. We show that, as a \(\mathrm {GL}_{n}(\mathbb {F}_q)\)-representation, this Jacquet module is isomorphic to \(\pi _{\theta \upharpoonright _{\mathbb {F}_n^*}} \otimes \mathrm {St}^{\otimes (k-1)}\), where \(\mathrm {St}\) is the Steinberg representation of \(\mathrm {GL}_{n}(\mathbb {F}_q)\). This generalizes a theorem of D. Prasad, who considered the case \(k=2\). We prove and rely heavily on a formidable identity involving q-hypergeometric series and linear algebra.

Notes

Acknowledgements

We are grateful to the second author’s advisor, David Soudry, for suggesting the problem and for many helpful discussions during our work on the case \(k=3\). We are thankful to Dipendra Prasad for interesting discussions. We are indebted to Dror Speiser for useful conversations, and in particular for suggesting the link with the Steinberg representation. We thank the referee for a careful reading of our manuscript and many comments and suggestions.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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