Xuhua He’s research while affiliated with The University of Hong Kong and other places

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 J -total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity. We show that the J J -totally nonnegative flag variety has a cellular decomposition into totally positive J J -Richardson varieties. Moreover, each totally positive J 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 J -Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J 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 in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of $U^{-}$ U − for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

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.

Motivated by Lusztig’s G-stable pieces, we consider the combinatorial pieces: the pairs (w, K) for elements w in the Weyl group and subsets K of simple reflections that are normalized by w. We generalize the notion of cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We show that the partial cyclic shift classes of combinatorial pieces associated with minimal-length elements have nice representatives. As applications, we prove the left-right symmetry and the compatibility of the induction functors of the parabolic character sheaves.

This 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 nonemptiness 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.

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 $G(\textbf{k})$ for any algebraically closed field $\textbf{k}$, which is an extension of the finite Weyl group $W_0$ 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.

Affine Lusztig varieties encode the orbital integrals of Iwahori-Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this paper, we establish a close relationship between affine Lusztig varieties and affine Deligne-Lusztig varieties. Consequently, we give an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases.

We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a regular CW complex homeomorphic to a closed ball. As a consequence, we show the totally nonnegative braid variety is a regular CW complex homeomorphic to a closed ball. Moreover, we prove that the link of any totally nonnegative double Bruhat cell in a reductive group is a regular CW complex homeomorphic to a closed ball, solving an open problem of Fomin and Zelevinsky.

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.

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.