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−1ℂ[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 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-Reshetikhin's and Talalaev-Chervov's constructions of higher Hamiltonians of the Gaudin model give the same family of commuting operators.