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

Goodwin, Simon Mark (2005)
Ph.D. thesis, University of Birmingham.

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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 \(M\).

Type of Work:Ph.D. thesis.
Supervisor(s):Rohrle, Gerhard
School/Faculty:Schools (1998 to 2008) > School of Mathematics & Statistics
Department:Mathematics and Statistics
Subjects:QA Mathematics
Institution:University of Birmingham
Library Catalogue:Check for printed version of this thesis
ID Code:118
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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