In this note we study simple modules for a reduced enveloping algebra Ux (g) in the critical case when X E 2 g^x is “nilpotent”. Some dimension formulas computed by Jantzen suggest modified versions of Weyl’s dimension formula, based on certain reflecting hyperplanes for the affine Weyl group which might be associated to Kazhdan–Lusztig cells.
Humphreys, J. E., "Analogues of Weyl’s Formula for Reduced Enveloping Algebras" (2002). Mathematics and Statistics Department Faculty Publication Series. 13.
Retrieved from http://scholarworks.umass.edu/math_faculty_pubs/13