Conjugacy classes in Sylow p-subgroups of finite Chevalley groups in bad characteristic

Abstract

Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less than or equal to 4, and $\mathbf G$ is not of type $F_4$. In these cases we deduce that $k(U)$ is given by a polynomial in $q$ with integer coefficients; this polynomial is different from the polynomial for good primes.