The Casimir connection of a simple Lie algebra g yields a family of commuting quantum Hamiltonians acting on the weight spaces of -modules. I will explain how to diagonalise these Hamiltonians by using the affine Kac--Moody algebra at critical level. This mirrors the construction of Feigin--Frenkel--Reshetikhin, who diagonalised the Gaudin Hamiltonians arising from the Knizhnik--Zamolodchikov connection, and leads to a new class of quantum integrable systems generalizing the Gaudin model. These integrable systems are quantisations of the shift of argument algebras, that is the maximal Poisson commutative subalgebras of introduced by Mishenko--Fomenko.
This is joint work with B. Feigin and E. Frenkel.