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Representations with minimal support for quantized Gieseker varieties

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Abstract

We study the minimally supported representations of quantizations of Gieseker moduli spaces. We relate them to \({\text {SL}}_n\)-equivariant D-modules on the nilpotent cone of \(\mathfrak {sl}_n\) and to minimally supported representations of type A rational Cherednik algebras. Our main result is character formulas for minimally supported representations of quantized Gieseker moduli spaces.

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Notes

  1. Note that in [18] minimally supported modules are labeled by \((n_0\lambda ,\varnothing ,\ldots ,\varnothing )\). The discrepancy here appears because there is a sign mistake in the proof of [18, Proposition 3.5] that leads to a reversal of the labeling. We fix the proof of [18, Proposition 3.5] in Proposition 3.4.

  2. Note that if m and n are not coprime, the coefficient in front of \(z^{m}\) in D(z) is not the character of a finite-dimensional representation and, in fact, does not need to be in \({\mathbb {Z}}[q, q^{-1}, q_{j}, q_{j}^{-1}\,|\,1 \le j \le r]\). This is the reason why we write “generating function” inside quotation marks.

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Acknowledgements

We would like to thank Eugene Gorsky and Monica Vazirani for useful discussions. We are also grateful to the anonymous referee for helpful comments that allowed us to improve the exposition. The work of P.E. was partially supported by the NSF under grant DMS-1502244. The work of V. K. was partially supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Pavel Etingof & Vasily Krylov

  2. Department of Mathematics National Research University Higher School of Economics, 6 Usacheva st., Moscow, 119048, Russian Federation

    Vasily Krylov

  3. Department of Mathematics, Yale University, 12 Hillhouse Avenue, New Haven, CT, 06511, USA

    Ivan Losev

  4. Department of Mathematics, University of California Davis, One Shields Avenue, Davis, CA, 95616, USA

    José Simental

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  1. Pavel Etingof
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  2. Vasily Krylov
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  3. Ivan Losev
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  4. José Simental
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Correspondence to José Simental.

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To the memory of Tom Nevins.

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Etingof, P., Krylov, V., Losev, I. et al. Representations with minimal support for quantized Gieseker varieties. Math. Z. 298, 1593–1621 (2021). https://doi.org/10.1007/s00209-020-02642-1

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