The Argument Shift Method and the Gaudin Model

We construct a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$. This family is parameterized by finite sequences $\mu$, $z_1,\dots,z_n$, where $\mu\in\mathfrak{g}^*$ and $z_i\in\mathbb{C}$. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For $n=1$, the corresponding commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional $\mathfrak{g}$-modules and the Gaudin model.