In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras.
An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism is nilpotent, that is, (ad X)n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X(t) is equal to tdim g. A semisimple Lie group or algebraic group G acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action.
matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group.
Geometrically, this orbit is a two-dimensional complex quadratic cone in four-dimensional vector space of
However, if we replace the complex special linear group with the real special linear group, new nilpotent orbits may arise.
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.
In the case of the special linear group SLn, 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 SLn, 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).