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Proceedings of the London Mathematical Society Advance Access published online on November 27, 2006
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdl009
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© 2006 London Mathematical Society

Good grading polytopes

Jonathan Brundan1 and Simon M. Goodwin2

1 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA brundan{at}darkwing.uoregon.edu
2 Institut for Matematiske Fag, Aarhus Universitet, DK-8000 Aarhus C, Denmark goodwin{at}maths.bham.ac.uk

Received 13 October 2005. Revision received 29 March 2006.

Let g be a finite-dimensional semisimple Lie algebra over C and eising a nilpotent element. Elashvili and Kac have recently classified all good Z-gradings for e. We instead consider good R-gradings, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of e in g. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good R-grading for e is independent of the particular choice of good grading.


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