A Kohno-Drinfeld theorem for quantum Weyl groups

TL;DR

This paper proves a conjecture relating a new flat connection's monodromy to quantum Weyl group representations for the Lie algebra sl_n, bridging geometric and algebraic approaches in quantum group theory.

Contribution

It establishes the equivalence between the monodromy of a novel flat connection and quantum Weyl group representations for sl_n, confirming a conjecture by Millson and the author.

Findings

Proves the conjecture for g=sl_n

Connects geometric monodromy with algebraic quantum group representations

Enhances understanding of quantum Weyl groups in Lie theory

Abstract

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}.