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

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 vs. Further let K be an extension field of the field of fractions of A and KH 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 KH-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 KH-module, this gives explicit formulae for basis elements belonging to a Wedderburn basis of KH. For the other left cells, similar relations are derived.These results in turn are used to find preimages of the standard basis elements tz 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 KH. 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.