Compositio Mathematica



Equivariant homology and K-theory of affine Grassmannians and Toda lattices


Roman Bezrukavnikov a1, Michael Finkelberg a2 and Ivan Mirkovic a3
a1 Department of Mathematics, Northwestern University, Evanston, IL 60208, USA bezrukav@math.northwestern.edu
a2 Independent Moscow University, 11 Bolshoj Vlasjevskij Pereulok, Moscow 119002, Russia fnklberg@mccme.ru
a3 Department of Mathematics, The University of Massachusetts, Amherst, MA 01003, USA mirkovic@math.umass.edu

Article author query
bezrukavnikov r   [Google Scholar] 
finkelberg m   [Google Scholar] 
mirkovic i   [Google Scholar] 
 

Abstract

For an almost simple complex algebraic group G with affine Grassmannian $\text{Gr}_G=G(\mathbb{C}(({\rm t})))/G(\mathbb{C}[[{\rm t}]])$, we consider the equivariant homology $H^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$ and K-theory $K^{G(\mathbb{C}[[{\rm t}]])}(\text{Gr}_G)$. They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of $G(\mathbb{C}[[{\rm t}]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin–Loktev fusion product of $G(\mathbb{C}[[{\rm t}]])$-modules.

(Received November 25 2003)
(Accepted May 26 2004)
(Published Online April 21 2005)


Key Words: affine Grassmannian; Toda lattice; Langlands dual group.

Maths Classification

19E08 (primary); 22E65; 37K10 (secondary).


Dedication:
Dedicated to Vladimir Drinfeld on the occasion of his 50th birthday