On the representations of disconnected reductive groups over F q

Let G be a connected reductive algebraic group defined over a finite field Fq. One of the main tools in the study of representations of the finite group G(Fq) over a field of characteristic zero is the use of certain varieties Xw (see [DL1]) on which G(Fq) acts (here w is a Weyl group element). [First paragraph] ©2012 Presented at the 2009-2011 Southeastern Lie Theory Workshop Series, held October 9-11, 2009 at North Carolina State University, May 22-24, 2010, at the University of Georgia, and June 1-4, 2011 at the University of Virgini