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

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 $u$.