The affine Springer fiber-sheaf correspondence

Eugene Gorsky; Oscar Kivinen; Alexei Oblomkov

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arXiv · 2204.00303

The affine Springer fiber-sheaf correspondence

Eugene Gorsky·Oscar Kivinen·Alexei Oblomkov

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Abstract

Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer theory and can be thought of as an incarnation of 3d mirror symmetry. For the group $GL_n$, the corresponding partial resolution is $\mathrm{Hilb}^n(\mathbb{C}^\times\times \mathbb{C})$. We also consider a quantization of this construction for homogeneous elements.

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Eugene Gorsky, Oscar Kivinen, Alexei Oblomkov. 2022-04-01. The affine Springer fiber-sheaf correspondence. https://arxiv.org/abs/2204.00303

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