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Unramified affine Springer fibers and isospectral Hilbert schemes

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Abstract

For any connected reductive group G over \({{\mathbb {C}}}\), we revisit Goresky–Kottwitz–MacPherson’s description of the torus equivariant Borel–Moore homology of affine Springer fibers \({\mathrm {Sp}}_{\gamma }\subset {{\,\mathrm{Gr}\,}}_G\), where \(\gamma =zt^d\) and z is a regular semisimple element in the Lie algebra of G. In the case \(G=GL_n\), we relate the equivariant cohomology of \({\mathrm {Sp}}_\gamma \) to Haiman’s work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of (n, dn)-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve \(\{x^n=y^{dn}\}\).

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Acknowledgements

I would like to thank Erik Carlsson, Victor Ginzburg, Eugene Gorsky, Mark Haiman, Matthew Hogancamp, Alexei Oblomkov, Hiraku Nakajima, Peng Shan, Eric Vasserot and Zhiwei Yun for interesting discussions on the subject. Special thanks to Roman Bezrukavnikov for suggesting the reference [18]. Additionally, I would like to thank the MSRI for hospitality, for most of the work was completed there during Spring 2018. This research was supported by the NSF grants DMS-1700814 and DMS-1559338, as well as the Vilho, Yrjö and Kalle Väisälä foundation of the Finnish Academy of Science and Letters.

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  1. Department of Mathematics, California Institute of Technology, Pasadena, USA

    Oscar Kivinen

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Kivinen, O. Unramified affine Springer fibers and isospectral Hilbert schemes. Sel. Math. New Ser. 26, 61 (2020). https://doi.org/10.1007/s00029-020-00587-1

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