Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen
- Cite this article as:
- Borho, W. & Kraft, H. Commentarii Mathematici Helvetici (1979) 54: 61. doi:10.1007/BF02566256
Let a reductive groupG act linearly on a vectorspaceV, and letO∋V be an orbit. For each irreducible representation ω ofG, the multiplicity with which ω occurs in the ring of regular functions onO (or on its closureŌ) provides an interesting numerical invariant ofO. We introduce an algebraic notion of a “deformation” of an orbit into another one. The main goal of this paper is to give sufficient conditions on an orbit, in order that an arbitrary deformation of this orbit has to preserve all the multiplicites mentioned above.
In the special case whereG=PSLn acts on its Lie-algebra in the usual way, every (nonzero) nilpotent orbit may be deformed into a semisimple orbit, and a conjecture of Dixmier amounts to saying that these deformations should preserve multiplicities. We prove this conjecture to be true in the case where the closure of the niloptent orbit is a normal variety.
In development of an idea of Dixmier , we also introduce a notion of “sheets” ofV (maximal irreducible subsets consisting of orbits of a fixed dimension), and we give a useful description (5.4) for all sheets of a semi-simple Lie-algebra.
This paper is strongly influenced by some investigations of B. Kostant  on similar problems and extends some of his results.