G. Lusztig
We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the other partition is defined using conjugation by a regular semisimple element.
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On some partitions of a flag manifold
在Kazhdan和作者的早期论文中，定义了从C上的约化连通群中的幂单类集合到外尔群中的共轭类的映射。在这里，我们提出了一些实验证据，可能是另一种定义的地图在某些分区的旗歧管的片。
A flag manifold over a semifield K can be partitioned into "half i-circles" which are orbits of a K-action on that flag manifold. Here i is fixed and it corresponds to a simple reflection in the Weyl group. We prove (for certain K) a connectedness property of the flag manifold: any two points of it can be joined by a finite sequence of half i-circles for various i.
For any semifield K we define a K-form of a partial flag manifold of a semisimple group G of simply laced type over the complex numbers. The definition is in terms of the theory of canonical bases.
Differential calculus on quantized irreducible flag manifolds
本文给出了量子不可约旗流形上微分运算的新描述，并定义了Dolbeault复形的量子类比，证明了其同调群与Andersen-Polo-Wen定义的一致。
Let G be a semisimple simply connected group of simply laced type over C. We show that the flag manifold of G has a version defined over the "tropical" semifield Z on which the monoid G(Z) associated to G and the semifield Z acts.
Remarks on totally positive flag manifolds
本文研究实数域上分裂半单群的全正部分与全正抛物子群的交，为全正旗流形提供新的注记。
Double affine Hecke algebras and affine flag manifolds, I
这堂课回顾了双仿射Hecke代数的单模的分类通过仿射型Steinberg簇的K-理论
A class of perverse sheaves on a partial flag manifold
在有限域上的连通约化群G的部分旗流形上引入了一类在G的有理点群作用下等变的反常层。这个类的定义类似于抛物特征层的定义。
The quantum group $\dot U$ and flag manifolds over the semifield Z
本文利用半域 Z 上旗流形，对修正量子群的典范基给出参数化，并修正了此前工作中的一些小错误。
Finite orbit decomposition of real flag manifolds
设G是连通实半单李群，H是闭连通子群.设P是G的极小抛物子群.证明了H在旗流形G/P上有开轨道当且仅当它在G/P上有2个开轨道，从而证实了T.松木
Categories of $D$-modules on a quantized flag manifold
本文针对量化旗流形上的 $D$-模范畴，比较 Lunts–Rosenberg 的 $\mathrm{Proj}$ 构造方法与 Backelin–Kremnizer 的等变 $D$-模方法。
Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant
给出了量子旗流形上量子微分算子层的整体瓣由量子群给出的定理的一个新的证明。作为推论，我们检索约瑟夫和Letzter的量子版本的经典包络代数定理的Duflo和Kostant。我们还刻画了量子群的ad-可积部分的中心以及伴随李代数在其上的作用。
Demazure crystal structure for flagged reverse plane partitions
本文研究带标志反向平面分拆，证明其集合可分解为 Demazure 晶体的不交并，并据此推出带标志对偶稳定 Grothendieck 多项式具有 key 正性。
On some partitions of an affine flag variety
本文讨论了仿射旗簇的几种划分。这些分区包括作为特殊情况下的分区仿射旗品种到仿射Deligne-Lusztig品种和仿射类似的分区旗品种到$\cb_w(b)$介绍的Lusztig在\cite{L1}的定义的一部分，字符层。其中，我们给出了一个公式的维数的仿射Deligne-Lusztig簇的经典循环群的阶的类多项式的扩展仿射Hecke代数。我们还证明了任何简单的$GL_n(\FF_q((\e)))$ -模作为仿射Deligne-Lusztig簇$X_w(1)$的上同调的子商出现，对于扩展的仿射外尔群$\ZZ^n \rtimes S_n$中的某些$w$，一定对于有限外尔群$S_n$中的某些$w$出现.对于$Sp_{2n}$也有类似的结果。
We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg.
Differential operators on quantized flag manifolds at roots of unity
量子化的旗流形是普通旗流形的$q$ -模拟，它被实现为一个非交换方案，我们可以利用非交换代数几何的框架定义其上的$D$ -模的范畴;然而，当参数$q$是单位根时，Lusztig的弗罗贝尼乌斯态射允许我们通过普通旗流形上的某个环层上的模来处理量子旗流形上的$D$ -模。在本文中，我们将证明，这层环的普通旗流形是一个Azumaya代数在其中心。我们还证明了它对某些子集的限制是分裂的Azumaya代数。这些结果类似于Bezrukavnikov-Mirković-Rumynin关于旗流形上正特征的$D$ -模的一些结果.
In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra $ C_{l}$ admit complex structures.
The quantum orbit method for generalized flag manifolds
具有泊松括号的广义旗流形是紧Poisson齐性空间，其在辛叶中的分解与其在Schubert胞腔中的分层一致.本文证明了相应的量子化旗流形的不可约 *-表示也是由它们的Schubert胞腔参数化的。一个重要的步骤是确定合适的代数生成的量化旗流形。这些代数生成元可以自然地用量子Plucker坐标表示。本说明补充了作者和Dijkhuizen在Commun的论文。Math.Phys.203（1999），pp. 297-324，其中这些结果是对广义旗流形的一个特殊子类建立的。
We first describe the tangent space to the affine flag manifold associated to a simple algebraic group over $\mathbb{C}$ at the distinguished point starting from standard definitions. We then construct projective lines in the affine flag manifold tangent to given root vectors associated to imaginary roots of the corresponding affine Kac-Moody algebra and describe in which Schubert varieties they lie.
Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups
我们猜想复旗流形G/B的适当的K-理论Gromov-Witten不变量是由根据Langlands-对偶量子化泛包络代数U_q（g '）构造的Toda系统的有限差分形式所支配的.在级数A的经典旗流形的情形下证明了该猜想。证明是基于一个完善的著名的Atiyah-Hirzebruch的论点适用于hyperquot-scheme紧空间的有理曲线在旗流形的算术亏格的刚性。
Invariant Einstein metrics on generalized flag manifolds with two isotropy summands
设$M=G/K$是一个广义旗流形，即紧半单李群$G$的伴随轨道.我们使用变分的方法找到不变的爱因斯坦度量的所有旗流形的两个各向同性和。我们还确定了这些爱因斯坦度量的性质作为临界点的数量曲率功能下固定的体积。
Alcove walks and nearby cycles on affine flag manifolds
利用Ram的凹室行走理论，给出了仿射Hecke代数的Bernstein表示的一个证明.该方法也适用于不等参数的情况。我们还讨论了如何帮助这些结果在研究层附近的周期仿射旗流形。
We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection between the sets of adjoint and coadjoint orbits of such groups. In addition, we show that the corresponding orbits, although different, are homotopy equivalent. We also provide a geometric description of the adjoint and coadjoint orbits of the Euclidean and orthogonal groups as a special class of flag manifold which we call a Hermitian flag manifold. These man…
The ring of differential operators on a quantized flag manifold
本文研究了量子旗流形上微分算子环的性质，并利用量子Weyl代数给出了其在仿射开子集上局部化的显式描述。
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 mani…
A remark on Schubert cells and duality of orbits on flag manifolds
已知旗流形上任意K_c-轨道的闭包表示为闭K_c-轨道与Schubert胞腔（[M2]，[Sp]）的乘积.我们已经将这一事实应用于旗流形（[GM]）上轨道的对偶性。我们在这里完善这个结果，并给出它的新的应用程序所产生的对偶域的研究。
Koppelman formulas on flag manifolds
对于值在全纯线丛以及重言式向量丛及其分支中的形式，我们构造了$\C^N$中旗流形上的Koppelman公式。作为应用，通过求解相应的$\debar$方程，得到了Bott-Borel-Weil型消灭定理的新的显式证明.我们还构造再生核的调和$(p,q)$形式的情况下，格拉斯曼。
Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds
我们研究了仿射旗流形上的某些邻近圈层，这些圈层是在仿射旗流形到仿射格拉斯曼流形的Beilinson-Gaitsgory变形中自然产生的。我们研究的多重性函数，我们介绍了在较早的文件中，它编码的数据的约旦-Hoelder系列。我们证明了重数函数是q中的多项式，并给出了它们的次数的一个精确界。我们的结果也适用于附近的周期在p-adic变形的Laumon-Haines-Ngo，也Wakimoto层。
We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold. We characterize when the generators with the same invariant geometry are in the same homotopy class. This is done by exploring the action of Weyl group on the irreducible components of isotropy representation of the flag manifold.
We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag manifolds to the thick case. As a consequence, we provide a description of the global sections of line bundles of a thick Schubert variety as conjectured in Kashiwara-Shimozono [Duke Math. J. 148 (2009)]. This also yields the existence of a compatible basis of thick Demazure modules, and the projective normality of the thick Schubert varieties.
Invariant Einstein metrics on flag manifolds with four isotropy summands
本文对具有4个迷向分量的广义旗流形进行分类，通过显式求解Einstein方程得到新的G-不变Einstein度量，并研究了这类度量的等距问题，丰富了齐性空间Einstein度量的研究成果。
The cohomology of the affine Deligne-Lusztig varieties in the affine flag manifold of $GL_2$
本文研究GL_2中仿射旗流形的Deligne-Lusztig簇。首先，我们确定所有这些品种同构。在此基础上，我们研究了群中元素b的σ-稳定子在仿射Deligne-Lusztig簇X_w（b）的etale上同调上的表示.我们描述这样的表示作为诱导从紧子群和noncuspidal表示。
Invariant Spinors on Flag Manifolds
本文基于正根集合的组合性质，刻画了复单李代数相关的极大旗流形上非平凡不变旋量的存在性，并给出了各情况下不变旋量空间维数的界。
An analogue of Weyl's law for quantized irreducible generalized flag manifolds
我们证明了量子化不可约广义旗流形的Weyl定律的一个类似物。我们的意思是定义一个zeta函数，类似于经典的设置，并表明它满足以下两个性质：作为量子化代数上的泛函，它与Haar态成比例;它的第一个奇点与经典维数一致。在更一般的紧量子群的情况下，给出了相应的公式.
We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivar…
Borel-type presentation of the torus-equivariant quantum $K$-ring of flag manifolds of type $C$
本文给出了$C$型旗流形的环面等变（小）量子$K$-环的Borel型展示。
We continue the study, begun in [Kouno-Naito-Orr-Sagaki, 2021], of inverse Chevalley formulas for the equivariant $K$-group of semi-infinite flag manifolds. Using the language of alcove paths, we reformulate and extend our combinatorial inverse Chevalley formula to arbitrary weights in all simply-laced types (conjecturally also for $E_8$).
Quantum $K$-theoretic Whitney relations for type $C$ flag manifolds
本文研究了类型C旗 manifold的量子K环的Whitney型呈现，通过半无限旗 manifold方法和Borel型呈现，给出了量子K环的完整定义关系。
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