For a compact topological space X with a continuous action of a compact topological group M’, the equivariant K-theory \(K_{M'}(X)\) is defined as the Grothendieck group of the category whose objects are the M’- equivariant complex vector bundles on X and the morphisms are M’- equivariant maps with locally constant rank, cf. G. Segal [ Inst. Haut. Étud. Sci., Publ. Math. 34, 129-151 (1968; Zbl 0199.262)]. Replacing M’ by a complex Lie group M underlying a reductive complex algebraic group, the author defines \(K_ M(X)\) and applies it to \(X=G/B\), where G is a simple, simply connected complex algebraic group and B is a Borel subgroup of G and to \(M=G\times {\mathbb{C}}^*\), where G acts on X by left translation on G and \({\mathbb{C}}^*\) acts trivially. Then \(K_ M(X)=K_ G(X)\otimes {\mathbb{Z}}[q,q^{-1}]\) and it is naturally a \(R_ M\)-module, where \(R_ M=K_ M(point)\) is the representation ring of M.
Let W be the Weyl group of G with respect to a maximal torus in B; P the lattice of weights, the group structure will be written multiplicatively; \(\tilde W=W\cdot P\) the semi-direct product with P normal, \(\tilde W\) contains the affine Weyl group as a subgroup of finite index; \(\tilde H\) the Hecke algebra corresponding to \(\tilde W\) which is an algebra over \({\mathbb{Z}}[q,q^{-1}]\) with generators \(T_ s\) (s\(\in S\), the canonical generators of W) and \(\theta_ p\) (p\(\in P)\) subject to a set of relations [cf. the author, Trans. Am. Math. Soc. 277, 623-653 (1983; Zbl 0526.22015)]. Then, he shows that \(K_ M(X)\) has a structure of \(\tilde H-\)module which commutes with the \(R_ M\)-module structure on \(K_ M(X)\) and he states a conjecture on the structure which gives some q- analogs of the Springer representations of the affine Weyl groups.