Quantum K-theoretic geometric Satake: the case

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 and 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$K\operatorname{Conv}(Gr)$whose morphism spaces are given by the$G^{\vee }\times \mathbb{C}^{\times }$-equivariant algebraic K-theory of certain fibre products. We conjecture that$K\operatorname{Conv}(Gr)$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$G=\operatorname{SL}_{n}$. A key tool in our proof is the$\operatorname{SL}_{n}$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$\operatorname{SL}_{n}$spider describes the category of$U_{q}\mathfrak{sl}_{n}$-equivariant${\mathcal{O}}_{q}(\operatorname{SL}_{n})$-modules. Then we use quantum loop algebras to relate the annular$\operatorname{SL}_{n}$spider to$K\operatorname{Conv}(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.