Casimir Operators and Monodromy Representations of Generalised Braid Groups

Abstract

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 ∇_κ on h with values in any finite-dimensional g-module V and simple poles on the root hyperplanes. The corresponding monodromyre presentation of the braid group B_g of type g is a deformation of the action of (afinite extension of) W on V. The residues of ∇_κ are the Casimirs κ_α of the subalgebra ssl^α₂ ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊂= V under the κ_α implies that, for generic values of the parameter, the braid group B_g 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 = sl₃ but that this is not the case if g ≇ sl₂,sl₃. 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 Usl_n-modules for n≥4. Finally, we study their reducibility 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 and g₂.