  Let g be a complex, simple Lie algebra and t a Cartan subalgebra of g. A new
unitary, flat connection on t with values in any finite-dimensional g-module V
and simple poles along the root hyperplanes was recently introduced by J.
Millson and myself. This connection depends upon a complex parameter h and I
conjectured that its monodromy is equivalent to the quantum Weyl group
representation of the braid group of type g defined by Lusztig,
Kirillov-Reshetikhin and Soibelman via the quantum group U_{h}g. In this paper,
I prove this conjecture for g=sl_{n}.
