Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic
Wenzel, Christian
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https://hdl.handle.net/2142/22442
Description
Title
Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic
Author(s)
Wenzel, Christian
Issue Date
1990
Doctoral Committee Chair(s)
Haboush, William J.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\Phi$, and a set of simple roots $\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties.
In my thesis I generalize to subgroup-schemes containing B. They are group-schemes, but not necessarily varieties; their algebras of functions might have nilpotent elements, i.e. they might not be reduced. In my thesis I show that in characteristic p $>$ 0, there are infinitely many whenever G $\not=$ 1, I exhibit their structure, and I classify them.
I show that in characteristic p $>$ 3, the subgroup-schemes containing B correspond to $\tilde\Delta$, the set of all maps from $\Delta$ to $\rm I\!N \cup \{\infty\},$ in such a way that it extends the classical classification of parabolic subgroups in terms of subsets of $\Delta$. To each $\varphi$ there is a parabolic P$\sb\varphi$ with $\rm P\sb\varphi = U\sb\varphi\cdot P\sb{I(\varphi)},$ I$(\varphi) = \{\alpha\in\Delta\mid\varphi(\alpha)=\infty\}$, $\rm P\sb{I(\varphi)} = (P\sb\varphi)\sb{red}$ = Spec(K (P$\sb\varphi$) /nilrad), U$\sb\varphi$ being a certain local unipotent subgroup-scheme. In characteristic 2,3 the situation is more complicated.
Furthermore I give a construction of G/P also for non-reduced P. I show that G/P is a rational projective variety, whenever char (K) $>$ 3.
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