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On the endomorphism algebra of generalised Gelfand-Graev representations


Author: Matthew C. Clarke
Journal: Trans. Amer. Math. Soc. 364 (2012), 5509-5524
MSC (2010): Primary 20G40
Published electronically: April 25, 2012
MathSciNet review: 2931337
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Abstract: Let $ G$ be a connected reductive algebraic group defined over the finite field $ \mathbb{F}_q$, where $ q$ is a power of a good prime for $ G$, and let $ F$ denote the corresponding Frobenius endomorphism, so that $ G^F$ is a finite reductive group. Let $ u \in G^F$ be a unipotent element and let $ \Gamma _u$ be the associated generalised Gelfand-Graev representation of $ G^F$. Under the assumption that $ G$ has a connected centre, we show that the dimension of the endomorphism algebra of $ \Gamma _u$ is a polynomial in $ q$, with degree given by $ \dim C_G(u)$. When the centre of $ G$ is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of $ q$, unless one adopts a convention of considering separately various congruence classes of $ q$. Subject to such a convention we extend our result.


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Additional Information

Matthew C. Clarke
Affiliation: Department of Mathematics, Trinity College, Cambridge, CB2 1TQ, United Kingdom
Email: matt.clarke@cantab.net

DOI: https://doi.org/10.1090/S0002-9947-2012-05543-7
Received by editor(s): September 21, 2010
Received by editor(s) in revised form: January 11, 2011
Published electronically: April 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.