The symmetric algebra $S(g)$ of a Lie algebra g carries a natural Poisson bracket. Shift of argument subalgebras (introduced by Fomenko and Mishchenko in 1978) form a family of maximal Poisson-commutative subalgebras in
$S(g)$ for semisimple $g$. This family is parametrized by regular elements of the dual space $g*$. I will discuss the quantization problem for shift of argument subalgebras, namely, how to lift these subalgebras to commutative subalgebras in the universal enveloping algebra $U(g)$, and how to describe the spectra of the quantum shift of argument subalgebras of $U(g)$ on (finite-dimensional) $g$-modules. These questions are related to the classical representation theory, in particular, it was observed by Vinberg, that the Gelfand-Tsetlin subalgebra in $U(gl_n)$ is a certain limit of quantum shift of argument subalgebras, and hence the spectra of quantum shift of argument subalgebras on a finite-dimensional $g$-module can be regarded as a deformation of the corresponding Gelfand-Tsetlin polytope.
The construction of the quantum shift of argument subalgebras is a version of the Feigin-Frenkel-Reshetikhin construction of higher hamiltonians for the Gaudin model. The quantum shift of argument subalgebras come from the center of the universal enveloping algebra of the corresponding affine Lie algebra $g^$ at the critical level by an appropriate quantum Hamiltonian reduction. The center at the critical level is naturally identified with the algebra of polynomial functions on the space of opers on the formal punctured disk with respect to the Langlands dual group $G^L$ (roughly, the space of gauge equivalence classes of connections in a principal $G^L$-bundle with some transversality condition). This allows us to treat the spectra of the quantum shift of argument subalgebras on $g$-modules as some subsets in the space of opers. I will give a precise description of these subsets for irreducible finite-dimensional g-modules.