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Generalized Affine Springer Theory and Hilbert Schemes on Planar Curves

Garner, Niklas and Kivinen, Oscar (2022) Generalized Affine Springer Theory and Hilbert Schemes on Planar Curves. International Mathematics Research Notices . ISSN 1073-7928. doi:10.1093/imrn/rnac038. (In Press)

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We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for GLₙ, as defined by Goresky–Kottwitz–MacPherson. Using a generalization of affine Springer theory for Braverman–Finkelberg–Nakajima’s Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn–Kamnitzer–Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp–Mellit, we also show the rational Cherednik algebra acts on the HOMFLY-PT homologies of torus knots. This work was inspired in part by a construction in 3D N = 4 gauge theory.

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Garner, Niklas0000-0002-4646-2599
Alternate Title:Hilbert schemes on plane curve singularities are generalized affine Springer fibers
Additional Information:© The Author(s) 2022. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model ( Received: 27 May 2021; Revision received: 21 January 2022; Accepted: 26 January 2022; Published: 02 March 2022. The authors thank Tudor Dimofte and Eugene Gorsky for discussions that initiated this project as well as for comments and for urging us to publish our results. We also thank Justin Hilburn, Joel Kamnitzer, and Alex Weekes for sharing their preliminary results in [23], and José Simental Rodriguez and Minh-Tam Trinh for comments on a draft of this paper. N.G. would like to thank Ingmar Saberi and José Simental Rodriguez for useful conversations. Part of this work was carried out during the KITP program Quantum Knot Invariants and Supersymmetric Gauge Theories (fall 2018), supported by NSF [grant PHY-1748958].
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Record Number:CaltechAUTHORS:20220425-287979900
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Official Citation:Niklas Garner, Oscar Kivinen, Generalized Affine Springer Theory and Hilbert Schemes on Planar Curves, International Mathematics Research Notices, 2022, rnac038,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:114449
Deposited By: Tony Diaz
Deposited On:25 Apr 2022 17:45
Last Modified:25 Apr 2022 17:45

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