Spherical nilpotent orbits and representations of semisimple Lie Groups: an overview
Donald King

Abstract: The vector space of complex symmetric nxn matrices is preserved by conjugation with complex nxn orthogonal matrices. Conjugacy classes (orbits) of height two nilpotent symmetric matrices have many pleasant properties, and give insights into the structure of interesting irreducible unitary representations of SL(n, R), the group of real nxn matrices of determinant one. If we replace SL(n, R) by a general reductive Lie group G, then its spherical nilpotent orbits have similar properties, and carry similar information about some of the irreducible unitary representations of G.