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Kazhdan-Lusztig basis, Wedderburn decomposition, and Lusztig’s homomorphism for Iwahori-Hecke algebras. (English) Zbl 1126.20002
Summary: Let \((W,S)\) be a finite Coxeter system and \(A:=\mathbb{Z}[\Gamma]\) be the group algebra of a finitely generated free Abelian group \(\Gamma\). Let \(\mathcal H\) 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 \(K\mathcal H\) 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 give a direct sum decomposition of the right regular \(K\mathcal H\)-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 \(K\mathcal H\)-module, this gives explicit formulae for basis elements belonging to a Wedderburn basis of \(K\mathcal H\). 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 \(\mathcal J\) under the Lusztig homomorphism from \(\mathcal H\) into the asymptotic algebra \(\mathcal J\). 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 \(\mathcal H\) into an \(A\)-subalgebra \(\mathcal L\) of \(K\mathcal H\). In the case that all left cell modules are simple (like, for example, in type \(A\)), \(\mathcal L\) is isomorphic to a direct sum of full matrix rings over \(A\).

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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