Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type
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- by Simon M. Goodwin and Gerhard Röhrle PDF
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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
- MR Author ID: 734259
- Email: goodwin@maths.bham.ac.uk
- Gerhard Röhrle
- Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
- MR Author ID: 329365
- Email: gerhard.roehrle@rub.de
- Received by editor(s): March 6, 2006
- Received by editor(s) in revised form: November 7, 2006
- Published electronically: July 30, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 177-206
- MSC (2000): Primary 20G40, 20E45; Secondary 20D15, 20D20
- DOI: https://doi.org/10.1090/S0002-9947-08-04442-5
- MathSciNet review: 2439403
Dedicated: Dedicated to Professor J. A. Green on the occasion of his 80th birthday