Operations on universal enveloping algebra and the "argument shift" method

If a vector field X is given on a Poisson manifold M such that the square of the Lie derivative in the X direction "kills" the Poisson bivector, then there is a well-known simple method of "shifting the argument" (along X) to construct a commutative subalgebra (with respect to the Poisson bracket) inside the algebra of functions on M. In a particular case, this method can be applied to the Poisson-Lie bracket on the symmetric algebra of an arbitrary Lie algebra and gives (according to a well-known result, the proven Mishchenko-Fomenko conjecture) maximal commutative subalgebras in the symmetric algebra. However, the lifting of these algebras to commutative subalgebras in the universal enveloping algebra, although possible, is based on very nontrivial results from the theory of infinite-dimensional Lie algebras. In my talk, I will describe partial results that allow one to construct on the universal enveloping algebra of the algebra gl_n the operators of "quasidifferentiation" and with their help, in some cases, construct a commutative subalgebra in Ugl_n. I will also describe how, in the general case, this question is reduced to the combinatorial question of commuting a certain set of operators in tensor powers R^n. The story is based on collaborations with Dmitry Gurevich, Pavel Saponov and Ikeda Yasushi.