``Lagrangian'' construction for representations of Hecke algebras
Let G be a complex semisimple Lie group and H the affine Hecke algebra over \({\mathbb{Z}}(q,q^{-1})\) associated to G. Motivated by a conjecture of Lusztig the author considers pairs (h,n), where \(h\in G\) is semisimple and n is a nilpotent element of the Lie algebra \({\mathfrak g}\) of G such that Ad h(n)\(=q_ 0^{-1}\cdot n\) for some \(q_ 0\in {\mathbb{C}}\), and associates (theorem 3) to such a pair a natural action of H on \(K^ i(X_{h,n})\). Here \(K^ i\) means the topological K-functor for \(i=0,1\) and \(X_{h,n}\) is the variety of all Borel subgroups containing both h and exp n. A key step in the construction of the natural action is the realization of H as an equivariant K-group (theorem 6). Similar results have been obtained by \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 80, 209-231 (1985; Zbl 0613.22003)]. The (positive) solution to several conjectures appearing in that paper was given by \textit{D. Kazhdan} and \textit{G. Lusztig} [ibid. 87, 153-215 (1987; Zbl 0613.22004)].
scientific article; zbMATH DE number 4015037
scientific article