Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type
| dc.creator | Goodwin, Simon M | |
| dc.creator | Roehrle, Gerhard E | |
| dc.date | 2006-02-01 | |
| dc.date | 2008-02-29 | |
| dc.date.accessioned | 2026-07-07T09:23:49Z | |
| dc.date.available | 2026-07-07T09:23:49Z | |
| dc.description | Let $G$ be a connected reductive algebraic group defined over the finite field $\FF_q$, where $q$ is a power of a good prime for $G$. We write $F$ for the Frobenius morphism of $G$ corresponding to the $\FF_q$-structure, so that $G^F$ is a finite group of Lie type. Let $P$ be an $F$-stable parabolic subgroup of $G$ and $U$ 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 in \cite{alperin}. In order to prove the result mentioned above, we consider, for unipotent $u \in G^F$, the variety $\CP^0_u$ of $G$-conjugates of $P$ whose unipotent radical contains $u$. We prove that the number of $\FF_q$-rational points of $\CP^0_u$ is given by a polynomial in $q$ with integer coefficients. Moreover, in case $G$ is split over $\FF_q$ and $u$ is split (in the sense of \cite[\S5]{shoji}), the coefficients of this polynomial are given by the Betti numbers of $\CP^0_u$. We also prove the analogous results for the variety $\CP_u$ consisting of conjugates of $P$ that contain $u$. | |
| dc.description | minor changes; to appear in Trans. Amer. Math. Soc | |
| dc.identifier | https://arxiv.org/abs/math/0602026 | |
| dc.identifier | http://arxiv.org/abs/math/0602026 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155875 | |
| dc.subject | Group Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 20G40, 20E45 | |
| dc.title | Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type | |
| dc.type | text |