Quantum Satake in type $A$. Part I

Abstract

We give an interpretation of \mathfrak {sl}_n -webs as morphisms between certain singular Soergel bimodules. We explain how this is a combinatorial, algebraic version of the geometric Satake equivalence (in type A ). We then q -deform the construction, giving an equivalence between representations of U_q (\mathfrak {sl}_n) and certain singular Soergel bimodules for a q -deformed Cartan matrix. In this paper, we discuss the general case but prove only the case n = 2, 3 . In the sequel we will prove n \geq 4 .