Traces on finite W-algebras

We compute the space of Poisson traces on a classical W-algebra modulo an arbitrary central character, i.e., linear functionals on such an algebra invariant under Hamiltonian derivations. 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-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 classical W-algebra modulo a central character.

💡 Research Summary

The paper investigates the relationship between Poisson traces on classical finite W‑algebras (modulo an arbitrary central character) and the topology of the corresponding Springer fibers. Starting with a nilpotent element e in a complex semisimple Lie algebra 𝔤, one constructs the classical W‑algebra Wχ via the Slodowy slice and Hamiltonian reduction. For a chosen central character λ, the authors consider the quotient Wχλ = Wχ / (Z – λ), where Z denotes the centre of Wχ. The main object of study is the space of Poisson traces \