  We prove that the $K$-$k$-Schur functions are part of a family of
inhomogenous symmetric functions whose top homogeneous components are Catalan
functions, the Euler characteristics of certain vector bundles on the flag
variety. 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 which conjecturally has both positive
branching and a rectangle factorization property. 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.
