Closed 𝑘-Schur Katalan functions as 𝐾-homology Schubert representatives of the affine Grassmannian

Recently, Blasiak–Morse–Seelinger introduced symmetric func- tions called Katalan functions, and proved that the K K -theoretic k 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 k -Schur Katalan functions is identified with the Schubert structure sheaves in the K K -homology of the affine Grassmannian. Our main result is a proof of this conjecture. We also study a K 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 K -theory ring of the flag variety to a closed K K - 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.