Representation Theory for Groups of Lie Type I

Explore the representation theory of finite groups of Lie type in this 52-minute lecture from the Hausdorff Trimester Program on Logic and Algorithms in Group Theory. Delve into the structural properties of these groups and their relevance to representation theory. Examine fundamental goals and current research in the field. Learn about Harish-Chandra theory as a crucial tool, and focus on Deligne-Lusztig theory for classifying irreducible complex representations. Cover topics including finite classical groups, exceptional groups, finite reductive groups, the Lang-Steinberg theorem, maximal tori, Weyl groups, and Coxeter groups. Gain insights into the classification of finite simple groups and the orders of various finite groups of Lie type.

Intro
THE CLASSIFICATION OF THE FINITE SIMPLE GROUPS
THE FINITE CLASSICAL GROUPS Examples for finite groups of Lie lype are the finite dassical
EXCEPTIONAL GROUPS
THE ORDERS OF SOME FINITE GROUPS OF LIE TYPE
FINITE GROUPS OF LIE TYPE VS. FINITE REDUCTIVE GROUPS
FINITE REDUCTIVE GROUPS Let G be a connected reductive algebraic group over Fandlet Fbe a Frobenius map of G
EXAMPLE: THE UNITARY GROUPS
THE LANG-STEINBERG THEOREM
MAXIMAL TORI AND THE WEYL GROUP
MAXIMAL TORI OF FINITE REDUCTIVE GROUPS
THE CLASSIFICATION OF MAXIMAL TORI
BN-PATRS
COXETER GROUPS
EXAMPLES FOR PARABOLIC SURGROUPS. IT