Nilpotent orbit

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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.

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[edit] Definition

Given a semisimple Lie group or algebraic group G, one considers those orbits in the adjoint representation on its Lie algebra whose Zariski closure contains 0. Such orbits are finite in number and are called (adjoint) nilpotent orbits of G.

[edit] Examples

Nilpotent 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 \lambda_1\geq \lambda_2\geq\ldots\geq\lambda_r, where λ 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,

A=\begin{bmatrix}x & y\\ z & -x \end{bmatrix}, \quad (x,y,z)\ne (0,0,0)\quad{\;} with x2 + yz = 0,

corresponding to the partition (2). Geometrically, this orbit is a two-dimensional complex quadratic cone in four dimensional vector space of 2\times 2 matrices minus its apex.

The complex special linear group is a subgroup of the general linear group with the same nilpotent orbits. However, if we replace the complex special linear group with the real special linear group, new nilpotent orbits may arise. In particular, for n=2 there are now 3 nilpotent orbits: the zero orbit and two real half-cones (without the apex), corresponding to positive and negative values of y − z in the parametrization above.

[edit] See also

[edit] References

  • David Collingwood and William McGovern. Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. ISBN 0-534-18834-6
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