Quantum K-theoretic geometric Satake

TL;DR

This paper develops a K-theoretic version of the geometric Satake correspondence involving quantum groups, providing a conjectural equivalence and proving it for the case of SL_n using combinatorial and diagrammatic methods.

Contribution

It introduces a K-theoretic convolution category related to quantum groups and proves the conjectured equivalence for SL_n using the $SL_n$ spider and quantum loop algebras.

Findings

Defined a convolution category $KConv(Gr)$ with equivariant algebraic K-theory.

Conjectured an equivalence between $KConv(Gr)$ and a category of $U_q\mathfrak{g}$-modules.

Proved the conjecture for $G=SL_n$ using combinatorial and diagrammatic techniques.

Abstract

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov-Finkelberg developed a derived version of this equivalence which relates the derived category of $ G^\vee$-equivariant constructible sheaves on $ Gr $ with the category of $G$-equivariant ${\mathcal O}(\mathfrak g)$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $ U_q \mathfrak g $. We define a convolution category $ KConv(Gr) $ whose morphism spaces are given by the $ G^\vee \times \mathbb C^\times $-equivariant algebraic K-theory of certain fibre products. We conjecture that $KConv(Gr)$ is equivalent to a full subcategory of the category of $ U_q \mathfrak g $-equivariant $ \mathcal O_q(G) $-modules. We prove this conjecture when $G = SL_n$. A key tool in our proof is the $SL_n$ spider, which is a combinatorial description of the category of $U_q \mathfrak{sl}_n$ representations. By applying horizontal trace, we show that the annular $SL_n$ spider describes the category of $ U_q \mathfrak{sl}_n $-equivariant $ \mathcal O_q(SL_n) $-modules. Then we use quantum loop algebras to relate the annular $SL_n $ spider to $ KConv(Gr) $. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.