Quantum affine algebras at roots of unity and equivariant K-theory

We show that the algebra homomorphism Uq(ĝln) ։ K GLd×C ∗ (Z)⊗C(q) constructed by Ginzburg and Vasserot between the quantum affine algebra of type gln and the equivariant K-theory group of the Steinberg variety (of incomplete flags), specializes to a surjective homomorphism U res ǫ (ĝln) ։ K GLd×C ∗ ǫ (Z). In particular, this shows that the parametrization of irreducible U res ǫ (ŝln)-modules and the multiplicity formulas of [7],[9] are still valid when ǫ is a root of unity. We will use the following notations: let n, d ∈ N. We set Vd = {(v1, v2 . . . vn) ∈ N n | ∑ i vi = d}; for v ∈ Vd put vi = v1 + . . .+ vi and identify v with the point ∑ i viei ∈ C n where (ei) is the canonical basis of C. For v, w ∈ Vd, let M(v, w) = { A = (aij) n i,j=1 ∈ N n | ∑