Equivariant K-theory of the partial flag varieties.

Abstract: Back in 1990 Beilinson, Lusztig, and MacPherson used convolution algebras of diagonal orbits in the double partial flag varieties over finite fields to provide a geometric framework for the quantum groups in type A. In 1998 Vasserot used equivariant K-theory of the Steinberg subvarieties in the cotangent bundle of the double partial flag varieties to provide a geometric framework for the affine quantum group.

In a joint project with Sergey Arkhipov, we define an algebra $\mathcal{A}_n$ that plays the role of a $q=0$ degeneration of the affine quantum group of type $A_n$, and use the equivariant K-theory of the double partial flag variety with $n$ steps to provide a geometric framework for it. Our algebra is defined via generators and relations. Then for each dimension $d$ of the ambient space, we show that there is a natural surjective map $\mathcal{A}_n\to A(n,d)$, were $A(n,d)$ is the equivariant K-theory of the double partial flag variety with n step in $\mathbb{C}^d$ equipped with the convolution product.

mathematical physicsalgebraic geometrycategory theoryrepresentation theory

Audience: researchers in the topic

UMass Amherst Representation theory seminar