Kirill Shakhmatov
In this paper we construct an equivariant embedding of the affine space $\mathbb{A}^n$ with the translation group action into a complete non-projective algebraic variety $X$ for all $n \geq 3$. The theory of toric varieties is used as the main tool for this construction. In the case of $n = 3$ we describe the orbit structure on the variety $X$.
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The goal of this paper is to prove a version of the non-abelian localization theorem for the rational equivariant K-theory of a smooth variety $X$ with the action of a linear algebraic group $G$. We then use this to prove a Riemann-Roch theorem which represents the completion of the higher equivariant K-theory of $X$ at various maximal ideals of the representation ring, in terms the equivariant higher Chow groups. This generalizes a result of Edidin and Graham to higher $K$-theory with rational coefficients.
Using the language of stacks, we recast and generalize a selection of results in equivariant geometry.
Let $G$ be an affine group scheme over a noetherian commutative ring $R$. We show that every $G$-equivariant vector bundle on an affine toric scheme over $R$ with $G$-action is extended from $\Spec(R)$ for several cases of $R$ and $G$. We show that given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant $K$-theories as well as equivariant $K'$-theories.
Thomason's completion for K-theory and cyclic homology of quotient stacks
本文证明了域上带群作用概形的等变K理论与循环同调的完备化定理，解决Thomason完备化问题，并给出有限稳定子群作用下等变同调的非等变描述及应用。
Equivariant completions of toric contraction morphisms
我们把环面压缩态射的等变完备化当作环面Mori理论的一个应用。为此，我们将复曲面Mori理论推广到非$\mathbb Q$ -阶乘复曲面簇.因此，我们的理论似乎是很大的不同，从里德的原始组合环面森理论。我们还解释了非$\mathbb Q$ -阶乘收缩的各种例子，这意味着$\mathbb Q$ -阶乘性在极小模型程序中起着重要的作用。这样，本文完成了复曲面Mori理论的基础，并向我们展示了极小模型程序的一个新方面。
In this paper, we study the algebraic analogue of the topological Atiyah-Segal completion theorem. We verify this completion theorem for the algebraic equivariant $K$-theory of smooth projective schemes. We also show that the completion theorem fails in general for smooth non-projective schemes.
In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K- Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorem…
Idempotent completions of equivariant matrix factorization categories
本文证明了与henselian局部超曲面环相关的等变矩阵分解范畴是幂等完备的，将Dyckerhoff在非等变情形下的结果推广至等变情形。
We describe a method to construct completions of affine spaces into total spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the projective line. As an application we provide families of examples with non-rational, birationally rigid and non-stably rational general fibers.
Equivariant embeddings into smooth toric varieties
我们刻画了代数簇到光滑复曲面簇和前簇的可嵌入性。我们的嵌入结果也保持在一个等变的上下文中，从而推广了一个著名的嵌入定理Sumihiro拟投射G-簇。其主要思想是将嵌入问题简化为仿射情形。这是通过构造等变仿射圆锥面来完成的，这是一种将射影G簇上的等变仿射圆锥的概念扩展到更一般框架的工具。
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the GKM condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the the formal affine Demazure algebra.
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials
研究了仿射旗流形关于Borel群作用的等变K-群。我们证明了K-群中（无限维）Schubert簇的结构层由唯一的多项式表示，我们称之为仿射Grothendieck多项式。
Algebraic cycles and completions of equivariant K-theory
设$G$是作用在代数空间$X$上的复线性代数群。本文证明了一个Riemann-Roch定理（定理5.3），该定理给出了等变Grothendieck群$G_0(G,X)$在表示环$R(G) \otimes \C$的任意极大理想上的等变圈的完备化.主要的新技术证明这个定理是我们的非阿贝尔完成定理（定理4.3）等变$K$理论。定理4.3将经典的可对角化群作用的局部化定理推广到任意群。
The Schubert bases of the torus-equivariant homology and cohomology rings of the affine Grassmannian of the special linear group are realized by new families of symmetric functions called k-double Schur functions and affine double Schur functions.
Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory
本文继解决Thomason的等变代数G-理论完备化猜想后，给出了非正则概形上等变完满复形同伦代数K-理论的导出完备化定理，适用广范围群与概形，还得到可对角化群下等变代数K-理论的对应定理及诸多应用。
We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties over algebraically closed fields of arbitrary characteristics) a 'finiteness' property of divisorial valuations over complex affine varieties proved…
Equivariant deformations of the affine multicone over a flag variety
我们证明了在一定的假设下，以旗簇上仿射多锥的等变变形为参数的不变Hilbert格式是一个仿射空间。证明是基于在一个固定的多投射空间中的一个奇妙的品种的建设。
Equivariant algebraic and semi-algebraic geometry of infinite affine space
本文研究了无限仿射空间中Sym(∞)作用下的代数与半代数几何性质，证明了n=1时的消去量词结果，但该结果在n>1时不成立。
We show how derived completion can be used to prove an analogue of Atiyah-Segal completion for the $T$-equivariant Hermitian K-theory of a scheme $X$ with a trivial $T$-action, containing $\tfrac{1}{2}$ and satisfying the resolution property, where $T$ is a split torus of rank $t$. This result is an important first step towards a more general Atiyah-Segal completion theorem for Hermitian K-theory.
Equivariant Vector Bundles on Certain Affine G-Varieties
利用线性代数数据，给出了某些仿射G簇（G是特征为0的代数闭域上的约化线性代数群）上的G等变向量丛范畴的具体刻画.
Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an étale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.
Equi-affine minimal-degree moving frames for polynomial curves
本研究提出仿射空间多项式曲线的极小次数等变活动标架构建理论与算法，给出将任意矩阵补全算法转化为等变活动标架算法的等变化流程并证明次数极小性，同时提出新型极小次数矩阵补全算法，补充相关领域研究。
Equivariant Algebraic K-Theory and Derived completions III: Applications
本文利用前序论文的导出完备化定理，在一般情形下建立了线性代数群作用下等变G-理论与等变同伦K-理论的Riemann-Roch定理，证明其适用于环面簇与球面簇，还得到Lefschetz-Riemann-Roch定理及相关计算谱序列。
On projective completions of affine varieties determined by 'degree-like' functions
本文研究了仿射代数簇在其正则函数环上由滤子或等价的“度样函数”给出的射影完备化。对于仿射簇的拟有限多项式映射P（即所有纤维都是有限的），我们证明了在P的无穷远处（即在对应于一般纤维的超曲面的完备化的交集中，并由P的分支函数确定），存在源的完备化。此外，我们表明，有“有限型”完成与后一个属性，确定了最大的有限数量的“半幂”，即映射的环的正则函数不包括零，到整数，其中发送产品到总和和总和到最大值（可能的例外时，summands有相同的半幂）。我们刻画了后一种类型的完备化的理想的超曲面在无穷远是激进的。此外，我们建立了一个一对一的对应关系之间的集合的最小相关联的素数的理想和唯一的最小集合的半度需要定义相应的度像函数。我们还证明了一个'仿射Bezout型'定理准有限多项式映射P承认半幂，使相应的完成不添加点在无穷远的P。
In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.
Equivariant vector bundles on group completions
本文讨论了具有正规交叉边界因子的约化代数群的双等变光滑（部分）紧化上的双等变向量丛范畴.我们的结果推广了A.A. Klyachko [数学。苏联。Izvestiya，{ \bf 35}，No.2（1990）]。作为应用，我们证明了伴随半单群在C意义下的奇妙紧化上的低秩等变向量丛的分裂。De Concini和C. Procesi [Lecture Note in Math. { \bf 996}（1983）].此外，我们还对B. Kostant在复杂群的情况下。
To smooth schemes equipped with a smooth affine group scheme action, we associate an equivariant motivic homotopy category. Underlying our construction is the choice of an `equivariant Nisnevich topology' induced by a complete, regular, and bounded $cd$-structure. We show equivariant $K$-theory of smooth schemes is represented in the equivariant motivic homotopy category. This is used to characterize equivariantly contractible smooth affine curves and equivariant vector bundles on such curves. Generalizations of the purity and blow-up theorems in motivic homotopy theory are shown for actions o…
Integral equivariant cohomology of affine Grassmannians
本文给出A型仿射格拉斯曼流形和旗流形的整等变上同调的显式表示，并将其与Lam和Shimozono关于有理等变上同调及Larson关于$\nmathbb{P}^1$向量丛模空间整上同调的结果进行比较。
Equivariant $K$-theory, affine Grassmannian and perfection
研究仿射Schubert变在完美仿射Grassmannian上torus-equivariant代数K-理论，并将其与完美复叠的torus-equivariantHochschild同调比较，证明在F_p线性下为同构，提出新的K-理论环计算方法。
We give a description of certain categories of equivariant coherent sheaves on Grothendieck's resolution in terms of the categorical affine Hecke algebra of Soergel. As an application, we deduce a relationship of these coherent sheaf categories to the categories of perverse sheaves considered in the work of Bezrukavnikov-Yun, generalizing results of Arkhipov-Bezrukavnikov. In addition, we deduce that the weak braid group action on sheaves of Riche and Bezrukavnikov-Riche can be upgraded to a strict braid group action.
Let G be a reductive group; in this note we give an interpretation of the dynamical Weyl group of of the Langlands dual group $\check{G}$ defined by Etingof and Varchenko in terms of the geometry of the affine Grassmannian Gr of G. In this interpretation the dynamical parameters of Etingof and Varchenko correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of affine Kac-Moody groups.
Riemann-Roch for equivariant Chow groups
本文证明了具有线性代数群$G$作用的概型或代数空间的等变Riemann-Roch定理。对于$G$ -空间$X$，该定理给出了等变Grothendieck群的完备化与等变等变Chow群的完备化之间的同构.证明这个同构的关键是等变Grothendieck群的完备化的几何描述。除了Riemann-Roch，这个结果还有一些纯$K$理论的应用。特别是，我们证明了一个猜想Köck（在情况下的定期计划）和扩展到任意特征的结果Segal表示环。
We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to the morphism lifting such pencil on a suitable resolution of its indeterminacies preserves the open subset P H â A^3. As an application, we obtain projective completions of A^3 into del Pezzo fibrations over P^1 of every degree less or equal to 4. We also obtain completions of A^3 into Mori conic bundles…
We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and give its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types.
Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces
设$X$是一个有理齐性空间，$QH^*(X)_{loc}^\times$是$X$的小量子上同调环中局部化在量子参数上的可逆元群.我们推广arXiv：math/0609796的结果，并明确实现arXiv：dg-ga/9511011中描述的映射$π_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$。我们还证明了这个映射是一个嵌入，并在等变量子上同调环$QH^*_T(X)_{loc}^\times$中实现了它。我们给出了这些元素的乘积的显式公式。证明依赖于一个推广，一个商的等变同调环的仿射格拉斯曼，一个公式证明了彼得马扎尔arXiv：0705.3826。它还使用彼得森的未发表的结果-最近证明的林和Shimozono在arXiv：0705.1386 -之间的比较等变同调环的仿射格拉斯曼和等变量子上同调环。
Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection si…
The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of C* whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant O2(C)-vector bundles.
We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $π_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of th…
Completions of the affine $3$-space into del Pezzo fibrations
本文构造了仿射3-空间到除7度外各度数del Pezzo纤维化全空间的完备化，刻画了可作闭纤维的del Pezzo曲面及全空间的奇点性质，并完整描述了诱导态射一般纤维的整曲线。
Completions of $\C^*$-surfaces
按照Dolgachev，Pinkham和Demazure的方法，我们在数学AG/0210153中对光滑仿射曲线上具有双曲$\Q$ -作用的法向仿射曲面进行了分类.在本文中，我们将展示如何从这种描述获得一个自然的等变完成这些$\C^*$ -曲面。利用初等变换，我们还推导出边界因子是数学AG/0511063意义上的标准图的自然完备化，并在某些情况下证明了它们的唯一性。这种描述在可由Z字形完成的法向仿射表面的情况下特别精确，即，由光滑的有理曲线组成的线性链。作为一个应用程序，我们分类的光滑或正常的$\C^*$ -曲面的边界出现的所有标记。
We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.
Equivariant intersection theory
本文发展了代数群在代数概型上作用的等变交理论。该理论是基于我们的建设等变周群。它们是等变上同调群的代数类似物，满足了普通Chow群的形式性质。此外，它们还具有等变上同调的许多性质。主要结果如下：（1）证明了光滑簇的几何同分块的Chow群上的标准交积的存在性--即使几何点的稳定子是非约化的。(2)构造了从相干层的等变$K$ -理论到等变Chow群的完备化的Todd类映射，证明了等变$K$ -理论的完备化与等变Chow群的完备化同构. (3)本文证明了环面作用的一个局部化定理，并利用它给出了环面作用在完备光滑簇上的Bott留数公式的特征自由证明。
On the geometry of SL(2)-equivariant flips
本文证明了任意三维正规仿射拟齐次SL（2）-簇都可以描述为四维仿射超曲面的范畴商。此外，我们还证明了任意三维正规仿射拟齐次SL（2）-簇的Cox环具有唯一的定义方程。这使得我们可以通过超曲面的不同GIT-映射来构造SL（2）-等变翻转。利用球面簇的理论，我们用二维色锥描述了SL（2）-翻转。
Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory
本文证明复数域上有限阿贝尔群等变动机谱范畴在素数完备化后等价于合成等变谱范畴的完备化，后者是等变谱的形变并范畴化等变完美偶滤过，其计算依赖于等变形式群律描述的等变代数配边同伦群。
Irreducible components of the equivariant punctual Hilbert schemes
设H_{ab}是在一维环面T_{ab}：={（t^{-b}，t^a），t \in k^*}的自然作用下参数化仿射平面不变量的0维子格式的等变希尔伯特格式。我们计算H_{ab}的不可约分量：它们与一组Hilbert函数一一对应。作为证明的副产品，我们给出了Ellingsrud和Stromme的结果的新证明，即计算Hilbert方案H^l的贝蒂数的主要引理，该方案参数化长度为l的仿射平面的0维子方案，以及通过显式平坦族描述H^l上的Bialynicki-Birula胞。特别是，我们精确的条件下的应用程序的这最后的描述。
Equivariant $K$-theory ring of the affine Grassmannian as deformation to normal cone
本文通过不动点局部化技术，将仿射格拉斯曼流形的等变$K$-理论环描述为仿射爆发的函数环，推广并证明了Bezukavnikov猜想，并给出无穷拓扑生成元。
On the canonical embeddings of certain homogeneous spaces
研究了齐性空间的等变仿射嵌入及其等变自同构。本文给出了一个拟仿射但非仿射的具有多个等变自同构的齐性空间的例子。本文证明了具有不动点和多个轨道的仿射嵌入的任何等变自同构连通群的可解性。这被应用于研究拟仿射齐性空间的代数幺半群和标准嵌入的轨道分解，即，与齐性空间的坐标代数相关联的仿射嵌入，只要后者的代数是双生成的。我们特别注意了模抛物子群的幂幺根的约化群的商空间的标准嵌入。对于这些品种，我们描述的轨道分解，计算模态的群作用，并找出它们是光滑的。我们还描述了这些规范嵌入的最小环境模块提供的作用群是单连通的。
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $\rightarrow$ X//Ga restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper Ga-actions, whose algebraic quotient morphisms $π$ :…
On the distribution of points of bounded height on equivariant compactifications of vector groups
证明了仿射空间的光滑等变紧化上有界高的有理点个数的渐近公式。(Nousétablissons un dépement asymptotique du nombre de points rationnels de hauteur schoolée sur les compactifications équivariallises de l'espace affine.）
Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.
In this article we continue from \cite{sub2-1} the study of normal analytic compactifications of $\cc^2$ from the point of view of their associated pencils of jets of curve germs centered at infinity. If $\bar X$ is a normal analytic compactification of $\cc^2$ which is {\em primitive}, i.e.\ $\bar X \setminus \cc^2$ is irreducible curve, then we show that $\bar X$ is projective iff $\bar X$ is algebraic iff at least one of the jets in the associated pencil of jets of curve-germs can be represented by a planar curve with one place at infinity. As a result we show that there are primitive norma…
We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $Γ$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $Γ$-admissible fans admit $Γ$-admissible completions. In order to show this we prove a combinatorial analog of noeth…
The equivariant K-theory of toric varieties
本文给出复曲面簇的等变K理论的两个结果。第一个是一个公式的等变K-群的任意仿射复曲面品种，推广已知的公式光滑的。事实上，这个结果是建立在一个更一般的背景下，涉及分次投射模的K理论。第二个结果是一个新的证明定理由于Vezzosi和Vistoli关于等变K理论顺利（不一定仿射）环面品种。
Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
对于半单连通代数群G，我们引入Nekrasov分拆函数的某些抛物类似物（由Nekrasov引入，最近由Nekrasov-Okounkov和Nakajima-Yoshioka研究，G=SL（n））。这些函数计算（粗略地说）射影平面上的主G-丛，在无穷远处具有平凡化，在水平线处具有抛物线结构。当上述抛物子群是博雷尔子群时，我们证明了相应的分拆函数基本上等于某个仿射李代数上的泛韦尔马模中的Whittaker矩阵系数，即其根系与李（G）的仿射化的根系对偶的李代数.我们解释了如何可以认为这个结果的仿射模拟结果的Givental和金的Gromov-Witten不变量（更准确地说，等变J函数）的旗流形。因此，本文的主要结果可以看作是通过相应的“Langlands对偶”仿射李代数来计算与G相关联的仿射旗流形的等变J函数（特别地，我们改进了通常旗流形的相应结果）。作为主要工具，我们使用代数几何版本的乌伦贝克空间介绍Finkelberg，Gaitsgory和作者。这些结果与塞伯格-威滕预势的联系将在以后的出版物中讨论。
The Leray-Hirsch Theorem for equivariant oriented cohomology of flag varieties
本研究借助形式仿射Demazure代数建立旗簇环面等变定向上同调的显式Leray-Hirsch定理，并将相关Borel模型推广至部分旗簇，拓展了等变上同调理论的适用范围。
Working over a ground field of characteristic zero, this paper studies the quotient morphism $π:X\to Y$ for an affine $\mathbb{G}_a$-variety $X$ with affine quotient $Y$. It is shown that the degree modules associated to the $\mathbb{G}_a$-action give a uniquely determined sequence of dominant $\mathbb{G}_a$-equivariant morphisms, $X=X_r\to X_{r-1}\to\cdots\to X_1\to X_0=Y$, where $X_i$ is an affine $\mathbb{G}_a$-variety and $X_{i+1}\to X_i$ is birational for each $i\ge 1$. This is the canonical factorization of $π$. We give an algorithm for finding the degree modules associated to the given …
Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions
本文研究仿射格拉斯曼流形的环面等变 $K$-同调环，在 $A$ 型中引入 $K$-理论双重 $k$-Schur 函数作为 Schubert 基，并给出 $\mathrm{Gr}_{{SL}_n}$ 的等变 $K$-同调环的 Ginzburg-Peterson 型实现。
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory.
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