Title:

Relative Springer isomorphisms and the conjugacy classes in Sylow psubgroups of Chevalley groups

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 \(M\).
