Traces on finite \mathcal{W} -algebras |
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Authors: | Pavel Etingof Travis Schedler |
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Affiliation: | 1. Department of Mathematics, MIT, Cambridge, MA, 02139, USA
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Abstract: | We compute the space of Poisson traces on a classical W \mathcal{W} -algebra, i.e., linear functionals invariant under Hamiltonian derivations. Modulo any central character, this space identifies
with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology
of the corresponding quantum W \mathcal{W} -algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that
the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology,
which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology
of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the centrally
reduced classical W \mathcal{W} -algebra. |
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