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Xuhua He

Publications and source records attributed to Xuhua He.

At least 37 records · Page 2Linked to original sources
arXiv2022

Affine Deligne--Lusztig varieties with finite Coxeter parts

In this paper, we study affine Deligne--Lusztig varieties $X_w(b)$ when the finite part of the element $w$ in the Iwahori--Weyl group is a partial $σ$-Coxeter element. We show that such $w$ is a cordial element and $X_w(b) \neq \emptyset$ if and only if $b$ satisfies a certain Hodge--Newton indecomposability condition. The main result of this paper is that for such $w$ and $b$, $X_w(b)$ has a simple geometric structure: the $σ$-centralizer of $b$ acts transitively on the set of irreducible components of $X_w(b)$; and each irreducible component is an iterated fibration over a classical Deligne--Lusztig variety of Coxeter type, and the iterated fibers are either $\mathbb A^1$ or $\mathbb G_m$.

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arXiv2022

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the $J$-totally nonnegative flag variety has a cellular decomposition into totally positive $J$-Richardson varieties. Moreover, each totally positive $J$-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive $J$-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam.

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arXiv2022

Total positivity and conjugacy classes

In this paper, we study the interaction between the totally positive monoid $G_{\ge 0}$ attached to a connected reductive group $G$ with a pinning and the conjugacy classes in $G$. In particular, we study how a conjugacy class meets the various cells of $G_{\ge0}$. We also state a conjectural Jordan decomposition for $G_{\ge0}$ and prove it in some special cases.

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arXiv2021

Demazure product of the affine Weyl groups

The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group $\tilde W$. The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell/shrunken Weyl chambers of $\tilde W$, and obtain an explicit formula on the Lusztig-Vogan map from the coweight lattice to the set of dominant coweights.

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arXiv2021

Stabilizers of irreducible components of affine Deligne--Lusztig varieties

We study the $J_b(F)$-action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of $J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.

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arXiv2021

Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras

Let $G$ be a connected reductive group over a non-archimedean local field $F$ and $I$ be an Iwahori subgroup of $G(F)$. Let $I_n$ is the $n$-th Moy-Prasad filtration subgroup of $I$. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with $I_n$-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups $G$ that split over an unramified extension of $F$. The first main result of this paper is a presentation of the Hecke algebra $\mathcal H(G(F),I_n)$, generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras. For split $GL_n$, Howe gave a refined presentation of the Hecke algebra $\mathcal H(G(F),I_n)$. To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group $W$ to $G(F)$. The study of a certain nice lifting of $W$ is the second main motivation of this paper, which we discuss below. In 1966, Tits introduced a certain subgroup of $G(\mathbf k)$, which is an extension of $W$ by an elementary abelian $2$-group. This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group. The "Tits group" $\mathcal T$ for the Iwahori-Weyl group $W$ is a certain subgroup of $G(F)$, which is an extension of the Iwahori-Weyl group $W$ by an elementary abelian $2$-group. The second main result of this paper is a construction of Tits group $\mathcal T$ for $W$ when $G$ splits over an unramified extension of $F$. As a consequence, we generalize Howe's presentation to such groups. We also show that when $G$ is ramified over $F$, such a group $\mathcal T$ of $W$ may not exist.

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arXiv2020

Basic loci of Coxeter type with arbitrary parahoric level

Motivated 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 [GH] and [GHN] 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.

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arXiv2020

A Birkhoff-Bruhat Atlas for partial flag varieties

A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the $J$-Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the $J$-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group. We also construct a combinatorial atlas for the index set $Q_K$ of the projected Richardson varieties in ${\mathcal {P}}_K$. As a consequence, we show that $Q_K$ has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams in the case where the group $G$ is a connected reductive group.

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arXiv2020

Dimension formula of the affine Deligne-Lusztig variety $X(μ, b)$

The study of certain union $X(μ, b)$ of affine Deligne-Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit formula of $\dim X(μ, b)$ for sufficiently large dominant coweight $μ$.

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arXiv2020

Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry.

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arXiv2020

Flag manifolds over semifields

In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that the flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac-Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch and others on the totally nonnegative flag manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the tropical flag manifolds (of finite type). As a by-product, we prove a conjecture of Lusztig on the duality of totally nonnegative flag manifold of finite type.

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arXiv2020

From conjugacy classes in the Weyl group to semisimple conjugacy classes

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple conjugacy classes of $G$. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.

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arXiv2020

Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas

We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

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arXiv2020

Cordial elements and dimensions of affine Deligne-Lusztig varieties

The affine Deligne-Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group $\mathbb G$ depends on two parameters: the $σ$-conjugacy class $[b]$ and the element $w$ in the Iwahori-Weyl group $\tilde W$ of $\mathbb G$. In this paper, for any given $σ$-conjugacy class $[b]$, we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde W$.

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arXiv2019

The m=2 amplituhedron

The (tree) amplituhedron $\mathcal{A}_{n, k, m}$ is introduced by Arkani-Hamed and Trnka in 2013 in the study of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is defined in terms of the totally nonnegative Grassmannians. In this paper, we show that the amplituhedron $\mathcal{A}_{n, k, m}$ for $m=2$ admits a triangulation. Our collection of cells is constructed via BCFW-type recursion. We also provide a diagrammatic interpretation of our construction.

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arXiv2019

Extremal cases of Rapoport-Zink spaces

We investigate qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

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arXiv2019

Fully Hodge-Newton decomposable Shimura varieties

The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties. We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic $σ$-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of \cite{HR}) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.

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