Uniqueness of higher Gaudin Hamiltonians

For any semisimple Lie algebra g , the universal enveloping algebra of the infinite-dimensional pro-nilpotent Lie algebra g _:= g ⊗ t − 1 ℂ [ t − 1 ] contains a large commutative subalgebra A ⊂ U ( g _ ) . This subalgebra comes from the center of the universal enveloping of the affine Kac-Moody algebra g ^ at the critical level by the AKS-scheme. In this note we show that the corresponding “classical” Poisson-commutative subalgebra gr A ⊂ S ( g _ ) is the Poisson centralizer of its simplest quadratic element, and deduce from this that the “quantum” subalgebra A ⊂ U ( g _ ) is uniquely determined by the classical one. As an application, we show that Feigin-Frenkel-Reshetikhinʹs and Talalaev-Chervovʹs constructions of higher Hamiltonians of the Gaudin model give the same family of commuting operators.