Casimir operators and monodromy representations of generalised braid groups
Let \(\mathfrak g\) be a complex simple Lie algebra with Cartan subalgebra \(\mathfrak h\) and Weyl group \(W\). In this paper the authors constructed a one-parameter family of flat connections \(\nabla_{\kappa}\) on \(\mathfrak h\) with values in any finite-dimensional \(\mathfrak g\)-module \(V\) and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group \(B_{\mathfrak g}\) of type \(\mathfrak g\) is a deformation of the action of \(W\) on \(V\). The residues of \(\nabla_{\kappa}\) are the Casimirs \(\kappa_{\alpha}\) of the subalgebras \({\mathfrak {sl}}_2^{\alpha}\subset \mathfrak g\) corresponding to the roots of \(\mathfrak g\). The irreducibility of a subspace \(U \subseteq V\) under the \(\kappa_{\alpha}\) implies that, for generic values of the parameter, the braid group \(B_{\mathfrak g}\) acts irreducibly on \(U\). They showed that these Casimirs act irreducibly on the weight spaces of all simple \(\mathfrak g\)-modules if \({\mathfrak {g=sl}}_3\) but that this is not the case if \({\mathfrak {g}}\ncong {\mathfrak{sl}}_2, {\mathfrak{sl}}_3\), which answered a question of Knutson and Procesi. Using this result, they 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 \(U_{\mathfrak h}{\mathfrak {sl}}_n\)-modules for \(n\geq4\). Finally, they studied the irreducibility of the action of the Casimirs on the zero-weight spaces of self-dual \(\mathfrak g\)-modules and obtained complete classification results for \({\mathfrak g}={\mathfrak {sl}}_n\) and \(\mathfrak g_2\).
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