Equivariant K-theory of the space of partial flags
Sergey Arkhipov, Mikhail Mazin
TL;DR
This work constructs a degenerating, Drinfeld-style affine $gl_n$-type algebra $\mathfrak{U}_n$ at $q=0$ and realizes its quotient as the affine $0$-Schur algebra ${\mathbb S}^{\operatorname{aff}}_0(n,d)$ via equivariant $K$-theory on varieties of partial flags. By translating Drinfeld generators into explicit, orbit-supported $K$-theory classes, the authors establish a surjection $\phi_d: \mathfrak{U}_n \to {\mathbb S}^{\operatorname{aff}}_0(n,d)$, with the image generated by diagonal and almost-diagonal orbit classes. They develop a robust combinatorial framework for orbit structures (via $n\times n$ matrices with nonnegative entries) and a detailed convolution calculus, including push-forwards/pull-backs and transversality, to control products of $K$-theory classes. A key contribution is the identification of local generators $\mathfrak{e}_{\mu,k}(p)$, $\mathfrak{f}_{\mu,k}(p)$, and $\mathfrak{h}_{\nu,n}(p)$ that replicate the essential relations of the degenerate affine quantum group and generate the whole convolution algebra. This work provides a q=0 affine Schur–Weyl-type correspondence in a geometric setting, with explicit algebra maps and generators that facilitate further study of degenerations and their applications in representation theory and algebraic geometry.
Abstract
We use Drinfeld style generators and relations to define an algebra $\mathfrak{U}_n$ which is a ``$q=0$'' version of the affine quantum group of $\mathfrak{gl}_n.$ We then use the convolution product on the equivariant $K$-theory of varieties of pairs of partial flags in a $d$-dimensional vector space $V$ to define affine $0$-Schur algebras ${\mathbb S}_0^{\operatorname{aff}}(n,d)$ and to prove that for every $d$ there exists a surjective homomorphism from $\mathfrak{U}_n$ to ${\mathbb S}_0^{\operatorname{aff}}(n,d).$