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
a2 Independent Moscow University, 11 Bolshoj Vlasjevskij Pereulok, Moscow 119002, Russia
a3 Department of Mathematics, The University of Massachusetts, Amherst, MA 01003, USA

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


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).

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