Nilpotent orbit

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 p_{ad X}(t) is equal to t^{dim 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. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent.

Nilpotent ${\displaystyle n\times n}$ matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group. From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \ldots \geq \lambda _{r},}$ where ${\displaystyle \lambda }$ is a partition of n. Thus in the case n=2 there are two nilpotent orbits, the zero orbit consisting of the zero matrix and corresponding to the partition (1,1) and the principal orbit consisting of all non-zero matrices A with zero trace and determinant,

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