Summary: We show that the algebra homomorphism \(U_q(\widehat{\mathfrak{gl}}_n)\twoheadrightarrow K^{\text{GL}_d\times\mathbf C^*}(Z)\otimes \mathbb{C}(q)\), constructed by Ginzburg and Vasserot between the quantum affine algebra of type \(\mathfrak{gl}_n\) and the equivariant \(K\)-theory group of the Steinberg variety \(Z\) (of incomplete flags), specializes to a surjective homomorphism \(U^{\text{res}}_\epsilon(\widehat{\mathfrak{gl}}_n)\twoheadrightarrow K_\epsilon^{\text{GL}_d\times\mathbf C^*}(Z)\). In particular, this shows that the parameterization of irreducible \(U^{\text{res}}_\epsilon(\widehat{\mathfrak{sl}})n\)-modules and the multiplicity formulas in [V. Ginzburg and É. Vasserot, Int. Math. Res. Not. 1993, No. 3, 67–85 (1993; Zbl 0785.17014); É. Vasserot, Transform. Groups 3, No. 3, 269–299 (1998; Zbl 0969.17009)] are still valid when \(\epsilon\) is a root of unity.