Symmetrisation and the Feigin-Frenkel centre

For complex simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the commutative subalgebra z ((g) over cap) subset of u(t(-1)g[t(-1)]) known as the Feigin-Frenkel centre. These formulas make use of the symmetrisation map as well as of some well-chosen symmetric invariants of g. There are some general results on the role of the symmetrisation map in the explicit description of the Feigin-Frenkel centre. Our method reduces questions about elements of z((g) over cap) to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G(2) by hand. One of our technical tools is the map m: S-k (g) -> Lambda(2) g circle times Sk-3 (g) introduced here. As the results show, a better understanding of this map will lead to a better understanding of z((g) over cap).