Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras

Based on a joint project with Alexander Molev.
Let $A_\mu$ be the shift of argument, also known as Mishchenko–Fomenko, subalgebra associated with a linear function $\mu$. Relying on the explicit description of the Feigin–Frenkel centre, we prove that the symmetrisation map solves Vinberg's quantisation problem in all classical types, assuming that $\mu$ is regular, and for any $\mu$ in types $A$ and $C$. Note that in type $A$ the results were known before. The symmetrisation map commutes with taking limits thus allowing one to quantise various limits of Mishchenko–Fomenko subalgebras.