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 $G = SL_n$. 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.