Abstract

The K-homology ring of the affine Grassmannian for the special linear group is realized as a subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars's relativistic Toda lattice with unipotent initial condition. We also discuss a correspondence between Schubert bases of these rings. This is a joint work with S. Iwao and T. Meano.