Skip to main content
SpringerLink
Log in

Traces on finite \( \mathcal{W} \)-algebras

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We compute the space of Poisson traces on a classical \( \mathcal{W} \)-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 \( \mathcal{W} \)-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 \( \mathcal{W} \)-algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.-L. Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114.

    MATH  MathSciNet  Google Scholar 

  2. C. Dodd, 2010, work in progress.

  3. P. Etingof, V. Ginzburg, Symplectic reection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Etingof, T. Schedler, Poisson traces and D-modules on Poisson varieties, arXiv:0908.3868, with an appendix by I. Losev, 2009.

  5. M. A. Farinati, A. Solotar, M. Suárez- Álvarez, Hochschild homology and cohomology of generalized Weyl algebras, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 2, 465–488.

    MATH  MathSciNet  Google Scholar 

  6. W. L. Gan, V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. (2002), no. 5, 243–255.

  7. R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), 327–358.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Losev, Finite W-algebras, arXiv:1003.5811, 2010.

  9. T. Levasseur, J. T. Stafford, Semi-simplicity of invariant holonomic systems on a reductive Lie algebra, Amer. J. Math. 119 (1997), 1095–1117.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Nest, B. Tsygan, Algebraic index theorem, Comm. Math. Phys. 172 (1995), no. 2, 223–262.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Slodowy, Four Lectures on Simple Groups and Singularities, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, Vol. 11, Rijksuniversiteit Utrecht Mathematical Institute, Utrecht, 1980.

  12. P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin, 1980.

  13. W. Soergel, The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 4, 535–538.

    MATH  MathSciNet  Google Scholar 

  14. T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Van den Bergh, Erratum to: “A relation between Hochschild homology and cohomology for Gorenstein rings" [Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348; MR1443171 (99m:16013)], Proc. Amer. Math. Soc. 130 (2002), no. 9, 2809–2810 (electronic).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Pavel Etingof & Travis Schedler

Authors
  1. Pavel Etingof
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Travis Schedler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel Etingof.

Additional information

Dedicated to Vladimir Morozov on the 100th anniversary of his birth

Rights and permissions

Reprints and permissions

About this article

Cite this article

Etingof, P., Schedler, T. Traces on finite \( \mathcal{W} \)-algebras. Transformation Groups 15, 843–850 (2010). https://doi.org/10.1007/s00031-010-9103-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-010-9103-8

Keywords

Search

Navigation