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Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic

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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
Doctoral Committee Chair(s): Haboush, W.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
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.
Issue Date: 1990
Type: Text
Language: English
URI: http://hdl.handle.net/2142/22442
Rights Information: Copyright 1990 Wenzel, Christian
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9114457
OCLC Identifier: (UMI)AAI9114457
 

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