K-theoretic Catalan Functions

The framework of Catalan functions provided new proof methods for resolving conjectures about k-Schur functions, which serve as Schubert representatives for the homology of the affine Grassmannian for SL_{k+1}. We prove that the K-theoretic refinement, K-k-Schur functions, are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for K-k-Schur functions produces a second shift-invariant basis satisfying a rectangle factorization property and that conjecturally has positive branching. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism. Finally, as a potential application to other affine settings, we provide conjectural descriptions for the Schubert homology representatives of the affine Grassmannian for Sp_{2n}.