We construct the Schubert basis of the torus-equivariant K-homology of the ane 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 where G = SLn, the K-homology of the ane Grassmannian is identied 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 ane 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 by means of Kashiwara’s thick ane ag manifold.
@cite_36: 
@cite_19: A snap action fluid control valve, the operation of which is controlled by a relatively slow acting thermally responsive actuator member. The valve of this invention is particularly adapted for use in controlling flow of fluid to a fluid operable clutch or the like for operation thereof. The thermally responsive actuator portion of the valve senses temperature of a fluid, the temperature of which is responsive to operation of a fan which is operated through the clutch.
@cite_14: The k-Young lattice Yk is a partial order on partitons with no part larger than k. This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103-146) from the study of the k-Schur functions sλ(k), Symmetric functions that from a natural basis of the space spanned by homogeneous funtions indexed by k-bounded partitions. The chains in the k-Young lattice are induced by a Pieritype rule experimentally satisfied by the k-Schur functions. Here, using a natural bijection between k-bounded partitons and k + 1-cores, we establish an algorithm for identifying chains in the k- Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric gruop S˜k+1 by a maximal parabolic subgruop. From this, the conjectured k-Pieri rule implies that the k-Kostka matrix connecting the homogeneous basis hλ λ∈Yk to sλ(k) λ∈Yk may now be obtained by counting appropriate classes of tableaux on k + 1-cores. This suggests that the conjecturally positive k-Schur expansion coefficients for Macdonald polynomials (reducing to q, t-Kostka polynomials for large k) could be described by a q, t-statistic on these tableaux, or equivalently on reduced words for affine permutations.
@cite_10: Abstract We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k -Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k - K -Schur functions – Schubert representatives for the K -theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
Morse @cite_10 gives a combinatorial definition of the affine stable Grothendieck polynomials @math in terms of affine set-valued tableaux and also proves the Pieri rule for @math . The original @math -Schur functions @math @cite_19 @cite_14 , which arose in the study of Macdonald polynomials, involve a parameter @math . It appears that a @math -analogue @math of @math exists , defined in a similar manner to [Conjecture 9.11] LLMS . The connection of @math and Macdonald theory is explored in @cite_36 .