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The based ring of the lowest two-sided cell of an affine Weyl group. II. (English) Zbl 0801.20016

The paper under review is the continuation of the author’s previous part I [J. Algebra 134, No. 2, 356-268 (1990; Zbl 0709.20021)]. Let \(H_{q_ 0}\) be the Hecke algebra over \(\mathbb{C}\) associated to an extended affine Weyl group \(W\) and to a parameter \(q_ 0 \in \mathbb{C}^*\). Let \(J\) be the asymptotic Hecke algebra defined by G. Lusztig [ibid. 109, 536-548 (1987; Zbl 0625.20032)]. Let \(K(J)\) (resp. \(K(H_{q_ 0})\)) be the Grothendieck group of \(J\)-modules (resp. \(H_{q_ 0}\)-modules) of finite dimension over \(\mathbb{C}\). Then the natural injective map \(\phi_{q_ 0}: H_{q_ 0} \to J\) induces a surjective homomorphism \((\phi_{q_ 0})_ *: K(J) \to K(H_{q_ 0})\). For each two-sided cell \(c\) of \(W\), we can define the direct summand \(K(J_ c)\) (resp. \(K(H_{q_ 0})_ c)\) of \(K(J)\) (resp. \(K(H_{q_ 0})\)). Thus \((\phi_{q_ 0})_ *\) induces a homomorphism \((\phi_{q_ 0})_{*,c}: K(J_ c) \to K(H_{q_ 0})_ c\). The map \((\phi_{q_ 0})_{*,c}\) remains surjective but is not always an isomorphism.
In the present paper, the author considers the case when \(c = c_ 0\) is the lowest two-sided cell of \(W\). By applying some of his previous results on the two-sided cell \(c_ 0\) and on the structure of the algebras \(H_{q_ 0}\) and \(J\) (see loc. cit.), the author makes a close investigation of the behavior of the map \((\phi_{q_ 0})_{*,c_ 0}\) on simple \(J_{c_ 0}\)-modules. Consequently, the author gives some conditions on the parameter \(q_ 0\) such that the map \((\phi_{q_ 0})_{*,c_ 0}\) is an isomorphism.

MSC:

20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20C33 Representations of finite groups of Lie type
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20E36 Automorphisms of infinite groups
18F30 Grothendieck groups (category-theoretic aspects)
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References:

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