Nilpotent Orbit - Poset Structure

Poset Structure

Nilpotent orbits form a partially ordered set: given two nilpotent orbits, O₁ is less than or equal to O₂ if O₁ is contained in the Zariski closure of O₂. This poset has a unique minimal element, zero orbit, and unique maximal element, the regular nilpotent orbit, but in general, it is not a graded poset. If the ground field is algebraically closed then the zero orbit is covered by a unique orbit, called the minimal orbit, and the regular orbit covers a unique orbit, called the subregular orbit.

In the case of the special linear group SL_n, the nilpotent orbits are parametrized by the partitions of n. By a theorem of Gerstenhaber, the ordering of the orbits corresponds to the dominance order on the partitions of n. Moreover, if G is an isometry group of a bilinear form, i.e. an orthogonal or symplectic subgroup of SL_n, then its nilpotent orbits are parametrized by partitions of n satisfying a certain parity condition and the corresponding poset structure is induced by the dominance order on all partitions (this is a nontrivial theorem, due to Gerstenhaber and Hesselink).