A Kohno-Drinfeld theorem for quantum Weyl groups

Abstract

Let $\mathfrak {g}$ be a complex, simple Lie algebra with Cartan subalgebra $\mathfrak {h}$ and Weyl group $W$. In [MTL], we introduced a new, $W$-equivariant flat connection on $\mathfrak {h}$ with simple poles along the root hyperplanes and values in any finite-dimensional $\mathfrak {g}$-module $V$. It was conjectured in [TL] that its monodromy is equivalent to the quantum Weyl group action of the generalised braid group of type $\mathfrak {g}$ on $V$ obtained by regarding the latter as a module over the quantum group $U\sb \hbar\mathfrak {g}$. In this paper, we prove this conjecture for $\mathfrak {g}=\mathfrak {sl}\sb n$.