New Directions in Representation Theory

The Iwahori?CHecke algebra H(G, B) of a finite Chevalley group G with respect to a Borel subgroup B is described as a deformation of the group algebra of the Weyl group of G Similarly, the +-part of the quantized enveloping algebra ${{U^+_v (\mathfrak{g})}}$ associated with a semisimple Lie algebra ${{\mathfrak{g}}}$ can be viewed as a deformation of the +-part of the universal enveloping algebra ${{U(\mathfrak{g})}}$ . In both cases it is shown how information concerning the deformed algebras H(G, B) and ${{U^+_v (\mathfrak{g})}}$ can be used to obtain results about the representation theory of the Chevalley group G and the semisimple Lie algebra ${{\mathfrak{g}}}$ .