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Affine Deligne–Lusztig varieties and quantum Bruhat graph

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Abstract

In this paper, we consider affine Deligne–Lusztig varieties \(X_w(b)\) and their certain union \(X(\mu ,b)\) inside the affine flag variety of a reductive group. Several important results in the study of affine Deligne–Lusztig varieties have been established under the so-called superregularity hypothesis. Such results include a description of generic Newton points in Iwahori double cosets of loop groups, covering relation in the associated Iwahori–Weyl group and a dimension formula for \(X(\mu ,b)\). We show that one can considerably weaken the superregularity hypothesis and sometimes eliminate it, thus strengthening these existing results.

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Notes

  1. For two elements \(w,w'\) of \(\widetilde{W}\), we say that \(w'\) is a cocover of w if w is a cover of \(w'\), i.e. \(w>w'\) and \(\ell (w')=\ell (w)-1\). We also shorthand these last two conditions by writitng \(w > rdot w'\).

  2. In fact, this is called depth in [2, 4] and [25]; we alter the terminology here to avoid any confusion with the notion of depth of a coweight defined earlier.

  3. In the same vein, we alter the terminology \(\ell _R\) used in loc. sit. for the definition of this length function and call it \(\ell _{\text {red}}\).

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Acknowledgements

I am grateful to my advisor Xuhua He for the suggestion to investigate these problems as well as numerous helpful discussions over the course of this work. I would also like to thank Jeffrey Adams and Thomas Haines for many helpful conversations and valuable feedback. Upon completion of writing this paper, I was informed that Felix Schremmer had independently obtained similar results. I thank him for his thoughtful comments on a preliminary draft of the paper, especially for pointing out a minor error in the manuscript.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Arghya Sadhukhan

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Correspondence to Arghya Sadhukhan.

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This work was partially supported by a Graduate School Summer Research Fellowship and NSF grant DMS-1801352 through Thomas Haines.

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Sadhukhan, A. Affine Deligne–Lusztig varieties and quantum Bruhat graph. Math. Z. 303, 21 (2023). https://doi.org/10.1007/s00209-022-03154-w

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