Quantizations of nilpotent orbits vs 1-dimensional representations of W-algebras

Details Quantizations of nilpotent orbits vs 1-dimensional representations of W-algebras Quantizations of nilpotent orbits vs 1-dimensional representations of W-algebras

2010-04-10

arxiv.org/abs/1004.1669v1

Mathematics

Representation Theory

16 pages

Scientific paper

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0 and O be a nilpotent orbit in g. Then Orb is a symplectic algebraic variety and one can ask whether it is possible to quantize $\Orb$ (in an appropriate sense) and, if so, how to classify the quantizations. On the other hand, for the pair (g,O) one can construct an associative algebra W called a (finite) W-algebra. The goal of this paper is to clarify a relationship between quantizations of O (and of its coverings) and 1-dimensional W-modules. In the first approximation, our result is that there is a one-to-one correspondence between the two. The result is not new: it was discovered (in a different form) by Moeglin in the 80's.

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