Comparison Of a-Functions

Abstract

. We show agreement of Lusztig's a-function with the a-function in [3]. 1. Introduction In this note, W denotes a finite, simply-laced Weyl group, or the affine Weyl group E 9 = E 8 . Let \Gamma g be the set of simple generators, which are parametrized by the nodes of the associated Coxeter graph \Gamma. Every w 2 W may be written as a product s 1 s 2 s 3 \Delta \Delta \Delta s n of simple generators. If n is minimal, we call this product "reduced" and define l(w) = n and supp(w) to be the set of generators which appear in the product. Let H be the Iwahori-Hecke algebra associated to W . This is an algebra over Q [q 1=2 ; q \Gamma1=2 ] with generators T s for each s 2 \Gamma g satisfying the relations T 2 s = (q \Gamma 1)T s + q, T s T t = T t T s if st = ts, and T s T t T s = T t T s T t if sts = tst, for distinct s, t 2 \Gamma g . Also, let v = q 1=2 . This algebra has a basis Tw , w 2 W , where we have Tw = T s1 \Delta \Delta \Delta T sn whenever s 1 \Delta \Delta \De..