Deligne Lusztig Varieties Research Articles

Rationality of extended unipotent characters

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof, we use realisations of characters in ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-adic cohomology groups of Deligne–Lusztig varieties as well as block theoretic considerations.

Newton strata in Levi subgroups

Certain Iwahori double cosets in the loop group of a reductive group, known under the names of P-alcoves or (J,w,δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(J,w,\\delta )$$\\end{document}-alcoves, play an important role in the study of affine Deligne–Lusztig varieties. For such an Iwahori double coset, its Newton stratification is related to the Newton stratification of an Iwahori double coset in a Levi subgroup. We study this relationship further, providing in particular a bijection between the occurring Newton strata. As an application, we prove a conjecture of Dong-Gyu Lim, giving a non-emptiness criterion for basic affine Deligne–Lusztig varieties.

Machine Learning Assisted Exploration for Affine Deligne–Lusztig Varieties

AbstractThis paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne–Lusztig varieties (ADLV). The primary objective is to investigate the non-emptiness pattern, dimension, and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.

Degrees of some orthogonal Deligne–Lusztig varieties

Degrees of some orthogonal Deligne–Lusztig varieties

Cohomology and geometry of Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}

We give a description of the cohomology groups of the structure sheaf on smooth compactifications X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} of Deligne–Lusztig varieties X(w) for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}, for all elements w in the Weyl group. As a consequence, we obtain the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}. Moreover, using our result for X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document} and a spectral sequence associated to a stratification of X¯(w)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{X}(w)$$\\end{document}, we deduce the modpm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{mod}\\ p^m$$\\end{document} and integral p-adic étale cohomology with compact support of X(w). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X(w), we show that a similar class of constructions provide smooth compactifications of X(w) in the case of GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document}. Furthermore, we show in the appendix that the Zariski closure of X(w), for any connected reductive group G and any w, has pseudo-rational singularities.

Semi-modules and crystal bases via affine Deligne-Lusztig varieties

There are two combinatorial ways of parameterizing the Jb(F)-orbits of the irreducible components of affine Deligne-Lusztig varieties for GLn and superbasic b. One way is to use the extended semi-modules introduced by Viehmann. The other way is to use the crystal bases introduced by Kashiwara and Lusztig. In this paper, we give an explicit correspondence between them using the crystal structure.

Affine Bruhat order and Demazure products

Abstract We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

Geometric $L$-packets of Howe-unramified toral supercuspidal representations

We show that L -packets of toral supercuspidal representations arising from unramified maximal tori of p -adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of L -packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toral L -packets by their characters on sufficiently regular elements of elliptic maximal tori.

Counting conjectures and e-local structures in finite reductive groups

We prove new results in generalised Harish-Chandra theory providing a description of the so-called Brauer–Lusztig blocks in terms of information encoded in the ℓ-adic cohomology of Deligne–Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the ℓ-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Broué, Fong and Srinivasan between ℓ-structures and their geometric counterpart. Finally, using the description of Brauer–Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisations of generalised Harish-Chandra series.

Affine Deligne–Lusztig varieties and folded galleries governed by chimneys

Affine Deligne–Lusztig varieties and folded galleries governed by chimneys

Connected components of affine Deligne–Lusztig varieties for unramified groups

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

On some simple geometric structure of affine Deligne–Lusztig varieties for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}_n$$\\end{document}

In this paper we study the geometric structure of affine Deligne–Lusztig varieties Xλ(b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\lambda }(b)$$\\end{document} for GLn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}_n$$\\end{document} and b basic. Motivated by Görtz and He (Camb J Math 3:323–353, 2015) and Chan and Ivanov (Math Ann 380:1801–1890, 2021), we consider a new condition on λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. If this is satisfied, then Xλ(b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_{\\lambda }(b)$$\\end{document} is the disjoint union of classical Deligne–Lusztig varieties times finite-dimensional affine spaces.

The Picard group of vertex affinoids in the first Drinfeld covering

AbstractLetFbe a finite extension of${\mathbb Q}_p$. Let$\Omega$be the Drinfeld upper half plane, and$\Sigma^1$the first Drinfeld covering of$\Omega$. We study the affinoid open subset$\Sigma^1_v$of$\Sigma^1$above a vertex of the Bruhat–Tits tree for$\text{GL}_2(F)$. Our main result is that$\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$, which we establish by showing that$\text{Pic}({\mathbf Y})[p] = 0$for${\mathbf Y}$the Deligne–Lusztig variety of$\text{SL}_2\!\left({\mathbb F}_q\right)$. One formal consequence is a description of the representation$H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$of$\text{GL}_2(\mathcal{O}_F)$as thep-adic completion of$\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.

Geometric structure of affine Deligne-Lusztig varieties for GL3

In this paper we study the geometric structure of affine Deligne-Lusztig varieties Xλ(b) for GL3 and b basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.

On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $\mathrm{GL}_n$

On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $\mathrm{GL}_n$

Affine Deligne–Lusztig varieties and quantum Bruhat graph

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.

COHOMOLOGY OF THE BRUHAT–TITS STRATA IN THE UNRAMIFIED UNITARY RAPOPORT–ZINK SPACE OF SIGNATURE

AbstractIn their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$ . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $ -adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$ . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.

Basic loci of Coxeter type with arbitrary parahoric level

AbstractMotivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.

Twisting operators and centralisers of Lie type groups over local rings

We extend the classical result asserting that the twisting operator preserves certain Deligne–Lusztig character values for truncated formal power series; along the way we discuss some properties of centralisers. Then, combining with the orbit construction, we apply it to analyse SL2 over finite dual numbers. In the appendix we discuss an action of GLn(Fq[[π]]/πr) on a Springer fibre intersected by Deligne–Lusztig varieties.

Irreducible components of affine Deligne–Lusztig varieties

By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to Miaofen Chen and Xinwen Zhu.