Quantum K-theoretic geometric Satake: the case

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.

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