CLOSED k-SCHUR KATALAN FUNCTIONS AS K-HOMOLOGY SCHUBERT REPRESENTATIVES OF THE AFFINE GRASSMANNIAN

Recently, Blasiak-Morse-Seelinger introduced symmetric functions called Katalan functions, and proved that the K-theoretic k-Schur functions due to Lam-Schilling-Shimozono form a subfamily of the Katalan functions. They conjectured that another subfamily of Katalan functions called closed k-Schur Katalan functions is identified with the Schubert structure sheaves in the K-homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K-theoretic Peterson isomorphism that Ikeda, Iwao, and Maeno constructed, in a nongeometric manner, based on the unipotent solution of the relativistic Toda lattice of Ruijsenaars. We prove that the map sends a Schubert class of the quantum K-theory ring of the flag variety to a closed K-k-Schur Katalan function up to an explicit factor related to a translation element with respect to an antidominant coroot. In fact, we prove this map coincides with a map whose existence was conjectured by Lam, Li, Mihalcea, Shimozono, and proved by Kato, and more recently by Chow and Leung.