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On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations

  • Alessandro PaoliniEmail author
  • Iulian I. Simion
Article

Abstract

Let G be a finite reductive group defined over \(\mathbb {F}_{q}\), with q a power of a prime p. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of G into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants with respect to a standard basis of the endomorphism algebra of a Gelfand-Graev representation of G when G = PGL3(q) for an arbitrary prime p, and when G = SO5(q) for p odd.

Keywords

Gelfand-Graev representations Kloosterman sums Refined Bruhat cells 

Mathematics Subject Classification (2010)

20C33 20C08 20F55 

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Notes

Acknowledgements

The authors deeply thank G. Malle for his precious comments and feedback on an earlier version of the paper. The authors are also grateful to the referee for their useful remarks. The first author acknowledges financial support from the SFB-TRR 195. Part of the work was developed during a research visit of the first author hosted at, and supported by, the Babeş-Bolyai University, and of the second author hosted at the Technische Universität Kaiserslautern and supported by the SFB–TRR 195. The authors would like to thank both institutions for the kind hospitality.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität KaiserslauternPostfachGermany
  2. 2.Department of MathematicsBabeş-Bolyai UniversityPloieştiRomania

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