Weights of KZ motives and quantum groups

Abstract

The Knizhnik--Zamolodchikov (KZ) connection is defined on the constant bundle determined by $n$ finite-dimensional irreducible representations of a simple Lie algebra $\mathfrak{g}$ and a nonzero complex parameter $κ$, over the configuration space of $n$ distinct points on the affine line. Work of Schechtman--Varchenko and Looijenga shows that, when $κ$ is rational, the resulting local systems admit natural motivic realizations, called KZ motives. The Kazhdan--Lusztig equivalence identifies the local systems underlying these motivic variations with certain representations of the pure braid group arising from the corresponding Lusztig quantum group at a root of unity. We formulate conjectures relating the weight filtrations of KZ motives to structures in quantum-group representation theory, particularly tilting modules and the Jantzen and Andersen filtrations, and provide evidence for these conjectures. Our methods use a new deformation of the representation ring that records the first moments of the weight filtration, together with Schmid's theorem on limiting mixed Hodge structures and results on the Hodge filtration obtained in earlier joint work with S. Mukhopadhyay.