Equivariant K-theory of the space of partial flags

TL;DR

This paper constructs a new algebraic framework using Drinfeld generators to study the equivariant K-theory of partial flag varieties, establishing connections with affine 0-Schur algebras and affine quantum groups.

Contribution

It introduces an algebra $rak{U}_n$ as a $q=0$ version of affine quantum groups and relates it to affine 0-Schur algebras via a surjective homomorphism.

Findings

Defined algebra $rak{U}_n$ using Drinfeld generators

Established a surjective homomorphism to affine 0-Schur algebras

Connected equivariant K-theory with quantum algebra structures

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 ${\mathbb S}_0^{\operatorname{aff}}(n,d).$