Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups

Goodwin, Simon Mark (2005). Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups. University of Birmingham. Ph.D.

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Abstract

Let $G$ be a simple linear algebraic group over the algebraically closed field $k$ . Assume $p$ = char $k$ > 0 is good for $G$ and that $G$ is defined and split over the prime field $\char{bbold10}{0x46}_p$ . For a power $q$ of $p$ , we write $G(q)$ for the Chevalley group consisting of the $\char{bbold10}{0x46}_q$ -rational points of $G$ . Let $F : G \rightarrow G$ be the standard Frobenius morphism such that $G^F$ = $G(q)$ . Let $B$ be an $F$ -stable Borel subgroup of $G$ ; write $U$ for the unipotent radical of $B$ and $\char{eufm10}{0x75}$ for its Lie algebra. We note that $U$ and $\char{eufm10}{0x75}$ are $F$ -stable and that $U(q)$ is a Sylow $p$ -subgroup of $G(q)$ . We study the adjoint orbits of $U$ and show that the conjugacy classes of $U(q)$ are in correspondence with the $F$ -stable adjoint orbits of $U$ . This allows us to deduce results about the conjugacy classes of $U(q)$ . We are also interested in the adjoint orbits of $B$ in $\char{eufm10}{0x75}$ and the $B(q)$ -conjugacy classes in $U(q)$ . In particular, we consider the question of when $B$ acts on a $B$ -submodule of $\char{eufm10}{0x75}$ with a Zariski dense orbit. For our study of the adjoint orbits of $U$ we require the existence of $B$ -equivariant isomorphisms of varieties $U/M \rightarrow$ $\char{eufm10}{0x75}$ / $\char{eufm10}{0x6d}$ , where $M$ is a unipotent normal subgroup of $B$ and $\char{eufm10}{0x6d}$ = Lie $M$ . We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all $M$ .