Xuhua He
We use $G$-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig's dimension estimates \cite[section 4]{L2}. This is a generalization of \cite{HL}.
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Hesselink strata in small characteristic and Lusztig-Xue pieces
证明单代数群$G$的幂零锥中卢斯蒂格引入的${\rm LX}(Δ)$片段与赫塞林克层$\frak{H}(Δ)$重合且构成划分，幂幺片段同理，关键验证特征2下${\rm E_7}$及特征2、3下${\rm E_8}$型群的划分性质。
We show that the definition of unipotent (resp. nilpotent) pieces for classical groups given by Lusztig (resp. Lusztig and the author) coincides with the combinatorial definition using closure relations on unipotent classes (resp. nilpotent orbits). Moreover we give a closed formula for a map from the set of unipotent classes (resp. nilpotent orbits) in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent (resp. nilpotent) pieces.
Let $G$ be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification $\bar{G}$ of $G$ into finite many $G$-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any $G$-stable piece in $\bar{G}$. We will show that the closure is a disjoint union of some $G$-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many $G$-orbits.
Lusztig's special pieces conjecture
研究简单代数群$G$的李代数$\mathfrak g$中特殊幂零轨道的特殊片，通过两种方法证明其是光滑$G$ - 簇被有限群作用的商，适用于经典$\mathfrak g$并给出新证，还涉及例外群及相关嵌入与定义。
In this notes, we will give an exposition of some results on the method of partial conjugation action. We first discuss the partial conjugation action of a parabolic subgroup of a Coxeter group. We then discuss some applications to Lusztig's $G$-stable pieces and its affine generalization. We also discuss some recent work on the $\s$-conjugacy classes of loop groups and affine Deligne-Lusztig varieties.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $\mathbf{k}$, and let Lie$(G)$ be its associated Lie algebra. In his series of papers on unipotent elements in small characteristic, Lusztig defined a partition of the unipotent variety of $G$. This partition is very useful when working with representations of $G$. Equivalently, one can consider certain subsets of the nilpotent variety of Lie$(G)$ called pieces. This approach appears in Lusztig's article from 2011. The pieces for the exceptional groups of type $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$ in bad charac…
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.
In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph.
In this paper we show that the tree class of a component of the stable Auslander-Reiten quiver of a Frobenius-Lusztig kernel is one of the three infinite Dynkin diagrams. For the special case of the small quantum group we show that the periodic components are homogeneous tubes and that the non-periodic components have shape Z[A_\infty] if the component contains a module for the infinite-dimensional quantum group.
We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}.
In this paper we show, using Deligne-Lusztig theory and Kawanaka's theory of generalised Gelfand-Graev representations, that the decomposition matrix of the special linear and unitary group in non defining characteristic can be made unitriangular with respect to a basic set that is stable under the action of automorphisms.
In this paper, we present a conjecture on the degree of unipotent characters in the cohomology of particular Deligne-Lusztig varieties for groups of Lie type, and derive consequences of it. These degrees are a necessary piece of data in the geometric version of Broué's abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Broué's conjecture should produce a more combinatorially defined derived equivalence, called a perverse equivalence. We prove that our conjectural degree is an integer (which is not obvious) and has the cor…
A greedy open-orbit criterion for solvable algebraic group actions, with applications to Lusztig's nilpotent varieties
该研究针对可解代数群作用提出贪心开轨道准则，应用于Dynkin箭图无重表示的Aut_Q(M)作用，简化Lusztig幂零簇刚性问题，为等定向A型提供显式算法。
Using a general result of Lusztig, we give explicit formulas for the dimensions of K^F-invariants in irreducible representations of G^F, when G=GL_n, F:G->G is a Frobenius map, and K is an F-stable subgroup of finite index in G^theta for some involution theta:G->G commuting with F. The proofs use some combinatorial facts about characters of symmetric groups.
We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.
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