In this paper, the authors construct some representation of the Hecke algebra of an affine Weyl group using equivariant K-theory and prove theorems close to the conjecture given by the second author in part I [Proc. Am. Math. Soc. 94, 337-342 (1985; Zbl 0571.22014)]. Let G be a reductive, simply connected complex Lie group; \(\alpha_ 0: \pi \to Z\) a locally trivial principal fibration with group G acting on \(\pi\) over the locally compact space Z; \(\alpha\) : \(G\to Z\) the associated bundle of groups, the fiber \({\mathbb{G}}_ z=\alpha^{-1}(z)\) is the quotient of \(\alpha_ 0^{-1}(z)\times G\) by the free G-action g: (p,g\({}_ 1)\to (gp,gg_ 1,g^{-1})\). Then, for any \(z\in Z\), \({\mathbb{G}}_ z\) is a complex Lie group isomorphic to G; the Weyl group \(({\mathbb{W}}_ z,{\mathbb{S}}_ z)\) of \({\mathbb{G}}_ z\) is canonically isomorphic to the Weyl group (W,S) of G.
Let \({\mathbb{B}}\) be the space of all Borel subgroups of the various \({\mathbb{G}}_ z\). Assume given a continuous section u of \(\alpha\) such that \(u_ z\) is a unipotent element of \({\mathbb{G}}_ z\) for all \(z\in Z\); a compact subgroup M of \(G\times {\mathbb{C}}^{\times}\) and \(\lambda\) : \(M\to {\mathbb{C}}^{\times}\) the character defined by \(\lambda ((g,a))=a\); continuous actions of M on \(\pi\), Z which are compatible with \(\alpha_ 0\) and the operation of G on \(\pi\), and for each \(m\in M\), \(z\in Z\), \(mu_ z=u_{mz}^{\lambda (m)}\). Then M acts naturally on \({\mathbb{B}}\) and it leaves stable the closed subspace \({\mathbb{B}}_ u\) of \({\mathbb{B}}\) consisting of all Borel subgroups of \({\mathbb{G}}_ z\) containing \(u_ z\) for various \(z\in Z\). One can define the equivariant K-theory with compact support \(K_ M({\mathbb{B}}_ u)\) which is an R(M)-module, R(M) being the representation ring of M.
Let P be an abelian group consisting of isomorphism classes of G- equivariant holomorphic line bundles on the variety of Borel subgroups of G. Now assume that G has no factors of type \(G_ 2\). For any \(s\in S\) and \(L\in P\), the authors define the endomorphisms \(T_ s\) and \(\theta_ L\) of \(K_ M({\mathbb{B}}_ u)\) and verify a modified form of the Iwahori-Matsumoto relations between them which make \(K_ M({\mathbb{B}}_ u)\) an \({\mathbb{H}}\otimes R(M)\)-module, where \({\mathbb{H}}\) is the abstract Hecke algebra over \({\mathbb{Z}}[q]\) (q an indeterminate). The construction provides a large number of square integrable representations of the Hecke algebra or of the algebraic group.