Quantum K-theoretic geometric Satake

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 $G=S{L}_n$. A key tool in our proof is the $S{L}_n$ 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 $S{L}_n$ spider describes the category of $U_q{\mathfrak{s}\mathfrak{l}}_n$-equivariant ${\mathcal{O}}_q(S{L}_n)$-modules. Then we use quantum loop algebras to relate the annular $S{L}_n$ spider to $KConv(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.