Algebraic Geometry and Representation Theory

0. Let G be a reductive group over Z. For any field F we can consider the group Gp of F-points on G. At first glance, the groups Gp for different fields F appear to have little in common with each other. I. Gelfand has conjectured that (1) The structure of the representations of GF has fundamental features which do not depend on a choice of F. (2) Moreover, it is possible to define representations by formulas which are universally valid over any local or finite fields. In the book [GGP-S] both conjectures are proved for G = SL ^ (see Chapter 2, §§4.1 and 5.4). Unfortunately the second conjecture is not known for any other group. Langlands reformulated the first conjecture in a more precise form [B], and it has been proven in a number of cases. Since almost nothing is known about the second conjecture, I will postpone its discussion until the very end of this paper and will restrict myself to applications of Algebraic Geometry to Representation Theory. 1. Finite fields. Let F be a finite field of characteristic p, G a reductive F-group, and G = GF- Let B = TQU C G be a Borei subgroup. For any character x of ïb, we denote by nx the induced representation 7rx = Ind^0 X« Such representations are called "the principal series representations. " We say that a character x is generic if xw ¥ " X f ° r anY nontrivial element w of the Weyl group W of G. It is easy to see that (1) for any generic character % • ^o — • C*, the induced representation 7rx is irreducible; (2) the representations 7rx, 7rx / are equivalent if and only if x>x ' a r e conjugate under the action of the Weyl group W; (3) for any generic t0 G T0 we have Tnrx(tQ) = T,wew x(*o)• Macdonald has conjectured that for any Cartan subgroup T C G and any generic character x: T — • C*, there exists a unique irreducible representation 7rx of G, such that for any generic t GT we have Tr nx(t) = ± J2wewT x(*™)> wher