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

### 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 $$M$$

Topics: QA Mathematics
Year: 2005
OAI identifier: oai:etheses.bham.ac.uk:118

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