Intersections of Deligne--Lusztig varieties and Springer fibres

Zhe Chen

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arXiv · 2101.00537

Intersections of Deligne--Lusztig varieties and Springer fibres

Zhe Chen

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Abstract

In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific Deligne--Lusztig variety; moreover, this component forms an open dense subset of a component of the Springer fibre. This boils down to a map from the unipotent variety to the Weyl group, and combines several constructions with a combinatorial flavour (like Weyr normal forms, Robinson--Schensted correspondence, and Spaltenstein's and Steinberg's labellings); it also provides a geometric interpretation of a classical dimension formula of unipotent centralisers.

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Zhe Chen. 2021-01-03. Intersections of Deligne--Lusztig varieties and Springer fibres. https://arxiv.org/abs/2101.00537

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