Casimir Operators and Monodromy Representations of Generalised Braid Groups
Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for generic values of the parameter, the braid group B acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g-modules if g is sl(3) but that this is not the case if g is not isomorphic to sl(2) or sl(3). We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin's braid group B_n on the zero weight spaces of all simple U_{q}sl(n)-modules for n greater or equal to 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self-dual g-modules and obtain complete classification results for g=sl(n) or g(2).