Representations of reduced enveloping algebras and cells in the affine Weyl group

Details Representations of reduced enveloping algebras and cells in the affine Weyl group Representations of reduced enveloping algebras and cells in the affine Weyl group

2005-02-04

arxiv.org/abs/math/0502100v2

Representations of algebraic groups, quantum groups, and Lie algebras, 63--72, Contemp. Math., 413, Amer. Math. Soc., Providen

Mathematics

Representation Theory

10 pages; LaTeX; submitted to AMS Contemporary Mathematics; minor revisions

Scientific paper

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p>0, and let g be its Lie algebra. The crucial Lie algebra representations to understand are those associated with the reduced enveloping algebra U_chi(g) for a nilpotent chi in g*. We conjecture that there is a natural assignment of simple modules in a regular block to left cells in the affine Weyl group (for the dual root system) lying in the two-sided cell which corresponds to the orbit of chi in Lusztig's bijection. This should respect the action of the component group of C_G(chi) and fit naturally into Lusztig's enriched bijection involving the characters of C_G(chi). Some evidence will be described in special cases.

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