K-theory Schubert calculus of the affine Grassmannian

2009/01/31 by Thomas Lam, Anne Schilling, Mark Shimozono · 1 citation
Mathematics · #math.CO #math.AG #msc:05E05 #msc:14N15

paper · pdf · doi:10.1112/s0010437x09004539

published as Compositio Math. 146 (2010) 811-852 · 38 pages

arxiv created 2009/08/31 · arxiv updated 2019/02/20

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.

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• Twisted factorial Grothendieck polynomials and equivariant K-theory of weighted Grassmann orbifolds