Spaltenstein, Jean Nicolas (1977) Dynkin varieties. PhD thesis, University of Warwick.

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Abstract

Let G be a linear algebraic group. The Dynkin variety ßx of an element x of G is the fixed point set of x on the variety ß of all Borel subgroups of G. We show that all irreducible components of this variety have the same dimension, and that ßx is connected if x is unipotent.
Suppose now that G is reductive (but not necessarily connected) and that x is unipotent. We generalize an inequality linking dim ßxand dim Zꓖ (x) and some results on the action of A₀(x) on the set S(x) of all irreducible components of ßx where A₀(x) is the group of components of ZGo(x). We consider also regular and sub-regular elements in non-connected reductive groups. For classical groups we get a combinatorial description for S(x) and the action of A₀(x) on S(x) and a formula for dim ßx We generalize to non-connected reductive groups a theorem of Richardson which associates to each conjugacy class of parabolic subgroups of G a unipotent class of G and for classical groups we get a combinatorial description of this map.
There is also some material on unipotent classes in arbitrary reductive groups.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Finite fields (Algebra), Lie groups, Geometry, Algebraic, Linear algebraic groups
Official Date:	September 1977
Dates:	Date Event September 1977 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Lusztig, George, 1946-
Sponsors:	Royal Society (Great Britain) ; Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Extent:	ix, 162 leaves
Language:	eng

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