Kazhdan–Lusztig basis, Wedderburn decomposition, and Lusztig's homomorphism for Iwahori–Hecke algebras

Abstract

Let ( W , S ) be a finite Coxeter system and A : = Z [ Γ ] be the group algebra of a finitely generated free abelian group Γ. Let H be an Iwahori–Hecke algebra of ( W , S ) over A with parameters v s . Further let K be an extension field of the field of fractions of A and K H be the extension of scalars. In this situation Kazhdan and Lusztig have defined their famous basis and the so-called left cell modules. In this paper, using the Kazhdan–Lusztig basis and its dual basis, formulae for a K-basis are derived that gives a direct sum decomposition of the right regular K H -module into right ideals each being isomorphic to the dual module of a left cell module. For those left cells, for which the corresponding left cell module is a simple K H -module, this gives explicit formulae for basis elements belonging to a Wedderburn basis of K H . For the other left cells, similar relations are derived. These results in turn are used to find preimages of the standard basis elements t z of Lusztig's asymptotic algebra J under the Lusztig homomorphism from H into the asymptotic algebra J . Again for those left cells, for which the corresponding left cell module is simple, explicit formulae for the preimages are given. These results shed a new light onto Lusztig's homomorphism interpreting it as an inclusion of H into an A-subalgebra L of K H . In the case that all left cell modules are simple (like for example in type A), L is isomorphic to a direct sum of full matrix rings over A.