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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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
Trans. Amer. Math. Soc. 361 (2009), 177-206 Request permission

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

  • Dedicated: Dedicated to Professor J. A. Green on the occasion of his 80th birthday
  • © 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