Algebraic Geometry and Representation Theory
نویسنده
چکیده
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.
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