Quantum K-theoretic geometric Satake: the SLn case

摘要

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(V)-equivariant constructible sheaves on Gr with the category of G-equivariant 0(g)modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group U(q)g. We define a convolution category KConv(Gr) whose morphism spaces are given by the G(V) x C-X-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)g-equivariant O-q(G)-modules. We prove this conjecture when G = SLn. A key tool in our proof is the SLn spider, which is a combinatorial description of the category of U(q)s(n) representations. By applying horizontal trace, we show that the annular SLn spider describes the category of U(q)sl(n)-equivariant O-q(SLn)- modules. Then we use quantum loop algebras to relate the annular SL, spider to KConv(Gr). This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.