Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups 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 \(M\).

Name dc:type.qualificationname

d_ph

Level dc:type.qualificationlevel

d_ph

Grantor dc:publisher.institution

University of Birmingham

Year dc:date.issued

2005

Author dc:creator

• Goodwin, Simon Mark

• QA Mathematics