On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem

Details On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem

2007-11-15

arxiv.org/abs/0711.2522v4

Mathematics

Representation Theory

final version; some minor corrections, including change of title (old title: "Remarks on Iwahori--Hecke algebras with unequal

Scientific paper

By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided that the Kazhdan--Lusztig basis of $H$ satisfies certain deep properties. If $W$ is crystallographic and $H$ is a one-parameter algebra, then these properties are known to hold thanks to a geometric interpretation. In this paper, we develop some new general methods for verifying these properties, and we do verify them for two-parameter algebras of type $I_2(m)$ and $F_4$ (where no geometric interpretation is available in general). Combined with previous work by Alvis, Bonnaf\'e, DuCloux, Iancu and the author, we can then extend Lusztig's construction of an explicit isomorphism to all types of $W$, without any restriction on the parameters of $H$.

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