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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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##### References:
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