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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type


Authors: Simon M. Goodwin and Gerhard Röhrle
Journal: Trans. Amer. Math. Soc. 361 (2009), 177-206
MSC (2000): Primary 20G40, 20E45; Secondary 20D15, 20D20
Published electronically: July 30, 2008
MathSciNet review: 2439403
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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$. We write $ F$ for the Frobenius morphism of $ G$ corresponding to the $ \mathbb{F}_q$-structure, so that $ G^F$ is a finite group of Lie type. Let $ P$ be an $ F$-stable parabolic subgroup of $ G$ and let $ U$ be the unipotent radical of $ P$. In this paper, we prove that the number of $ U^F$-conjugacy classes in $ G^F$ is given by a polynomial in $ q$, under the assumption that the centre of $ G$ is connected. This answers a question of J. Alperin (2006).

In order to prove the result mentioned above, we consider, for unipotent $ u \in G^F$, the variety $ \mathcal{P}^0_u$ of $ G$-conjugates of $ P$ whose unipotent radical contains $ u$. We prove that the number of $ \mathbb{F}_q$-rational points of $ \mathcal{P}^0_u$ is given by a polynomial in $ q$ with integer coefficients. Moreover, in case $ G$ is split over $ \mathbb{F}_q$ and $ u$ is split (in the sense of T. Shoji, 1987), the coefficients of this polynomial are given by the Betti numbers of $ \mathcal{P}^0_u$. We also prove the analogous results for the variety $ \mathcal{P}_u$ consisting of conjugates of $ P$ that contain $ u$.


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Additional Information

Simon M. Goodwin
Affiliation: School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
Email: goodwin@maths.bham.ac.uk

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Email: gerhard.roehrle@rub.de

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04442-5
PII: S 0002-9947(08)04442-5
Received by editor(s): March 6, 2006
Received by editor(s) in revised form: November 7, 2006
Published electronically: July 30, 2008
Dedicated: Dedicated to Professor J. A. Green on the occasion of his 80th birthday
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.