Compositio Mathematica

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Compositio Mathematica (2010), 146:811-852 London Mathematical Society
Copyright © Foundation Compositio Mathematica 2010
doi:10.1112/S0010437X09004539

Research Article

K-theory Schubert calculus of the affine Grassmannian


Thomas Lama1, Anne Schillinga2 and Mark Shimozonoa3

a1 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: tfylam@math.harvard.edu)
a2 Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA (email: anne@math.ucdavis.edu)
a3 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (email: mshimo@vt.edu)
Article author query
lam t [Google Scholar]
schilling a [Google Scholar]
shimozono m [Google Scholar]

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 where G=SLn, 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, calledK-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 by means of Kashiwara’s thick affine flag manifold.

(Received May 05 2009)

(Accepted August 31 2009)

(Online publication January 26 2010)

2000 Mathematics Subject Classification05E05; 14N15 (primary)

Keywordsaffine Grassmannian; K-theory; Schubert calculus; symmetric functions; GKM condition

Footnotes

This work was partially supported by the NSF grants DMS-0600677, DMS-0501101, DMS-0652641, DMS-0652648 and DMS-0652652.