On conjugacy of unipotent elements in finite groups of Lie type

TL;DR

This paper proves that the number of conjugacy classes of unipotent radicals in finite groups of Lie type can be expressed as a polynomial in the size of the underlying finite field.

Contribution

It establishes that the conjugacy class count for unipotent radicals in these groups is given by a polynomial in q with integer coefficients, a new algebraic insight.

Findings

Number of conjugacy classes is polynomial in q.

Polynomial has integer coefficients.

Applicable to groups over finite fields of good characteristic.

Abstract

Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficients.