Dimensions of affine Deligne-Lusztig varieties in affine flag varieties
Ulrich Goertz, Xuhua He

TL;DR
This paper investigates the non-emptiness and dimension of affine Deligne-Lusztig varieties in affine flag varieties, confirming conjectures for split classical groups and basic classes, advancing understanding of their geometric structure.
Contribution
It proves a conjecture on non-emptiness of affine Deligne-Lusztig varieties for split groups and basic classes, and verifies the dimension formula for classical groups in most cases.
Findings
Confirmed non-emptiness conjecture for affine Deligne-Lusztig varieties.
Validated the dimension formula for classical groups with trivial basic class.
Extended understanding of the geometric properties of affine Deligne-Lusztig varieties.
Abstract
Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic -conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic -conjugacy class is the class of , we also prove the dimension formula predicted in op. cit. in almost all cases.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
