K-theory Schubert calculus of the affine Grassmannian

TL;DR

This paper constructs a Schubert basis for the equivariant K-homology of the affine Grassmannian, linking algebraic and geometric structures, and introduces K-k-Schur functions and affine stable Grothendieck polynomials.

Contribution

It develops a K-theoretic Schubert calculus for the affine Grassmannian, including new bases, symmetric functions, and a Pieri rule, extending prior homology constructions.

Findings

Identification of K-homology with symmetric functions for SL_n

Introduction of K-k-Schur functions as Schubert basis

Verification of Lam's conjecture on affine stable Grothendieck polynomials

Abstract

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case G = SL_n, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called K-k-Schur functions, whose highest degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations using Kashiwara's thick affine flag manifold.