Nilpotent orbits, normality and Hamiltonian group actions

Abstract

The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncompact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semisimple groups. Here the associated representation theory (unipotent representations) is poorly understood and there is a loss of reductivity of isotropy subgroups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). This paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry. The starting point is the recognition (made also by others) that the ring R of regular functions on any G-homogeneous cover M of 0 is not only a Poisson algebra (the case for any coadjoint orbit) but that R is also naturally graded. The key theme is that the same nilpotent orbit may be by more than one simple group. The key result is the determination of all pairs of simple Lie groups having a shared nilpotent orbit. Furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic