Conjugacy Classes in Algebraic Groups (Lecture Notes in Mathematics) - ISBN:3540066578

Conjugacy Classes in Algebraic Groups (Lecture Notes in Mathematics)

Date: 09 April 2011, 14:44

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The following is the substance of a set of lectures given at the Tata Institute of Fundamental Research during November and December of 1972, The notes are divided roughly into two parts. The first part attempts an a priori development of the basic properties of affine algebraic groups with emphasis on those needed in the study of conjugacy classes of elements of reductive groups : the semi- simple-unipotent decomposition, conjugacy of Borel subgroups and of maximal tori, completeness of the variety of Borel subgroups, etc. The second part is devoted to the classification and characterization of various such classes of elements: semisimple, unipotent, regular, subregular, etc. For a more detailed outline the reader may consult the table of contents. All of this is over an algebraically closed field. I had planned to include two talks on rationality questions, but this aim was not realized. Because of time limitations there had to b gaps in the actual development. In the first part the most serious of these is the omission of a large part of the proof of the existence of a quotient of a group by a closed subgroup. Also the principal structural and conjugacy results about connected solvable groups are used without proof, but this is not so serious since the Lie-Kolchin theorem is proved and from there on the proofs, by induction, follow fairly classical lines. In the second part the Bruhat lemma for reductive groups is used without proof (but a fairly complete proof is indicated for the classical groups) as are various properties of root systems and reflection groups (for which a comprehensive treatment may be found in Bourbaki's book). Modulo a few other points left to be checked by the reader I have attempted a coherent development.

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