Anatolii V. Tushev
In the paper we show that any irreducible representation of a finitely generated nilpotent group $G$ over a finitely generated field $F$ of characteristic zero is induced from a primitive representation of some subgroup of $G$.
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Finite group algebras of nilpotent groups: a complete set of orthogonal primitive idempotents
给出了幂零群上有限群代数的正交本原幂等元完备集的一个显式构造。进一步，我们给出了幂零群的有限群代数的每个单满射像中的矩阵单位的完备集。
Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups
设$G$是一个无挠生成的幂零群. $G$的表示zeta函数$ζ_G(s)$列举了$G$的有限维不可约复表示的扭曲类。证明了$ζ_G(s)$具有有理收敛横坐标$a(G)$，并且可以亚纯地延拓到$a(G)$的左边，并且在$\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$线上，除了在$s=a(G)$上的极点外，该延拓函数是全纯的.一个Tauberian定理产生一个精确的渐近结果的表示增长的$G$的位置和顺序的极点。我们得到这些结果作为一个更一般的结果建立一致的分析性质的表示zeta函数的生成幂零群的形式$\mathbf{G}(\mathcal{O})$，其中$\mathbf{G}$是一个幂零群计划定义的幂零李格环$\mathcal{O}$的整数字段。这使我们能够显示，特别是，横坐标的收敛的代表zeta功能的这些群体和他们的极点阶是不变量的$\mathbf{G}$，独立的$\mathcal{O}$。
A result of D. Segal states that every complex irreducible representation of a finitely generated nilpotent group $G$ is monomial if and only if $G$ is abelian-by-finite. A conjecture of A. N. Parshin, recently proved affirmatively by I.V. Beloshapka and S. O. Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture combining the ideas of I. D. Brown and P. C. Kutzko. We also characterize finite dimensional irreducible representations of two step nilpotent groups and also pro…
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(λ)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits
本文研究了有限维单李代数中与刚性幂零轨道关联的无重数本原理想的数量，求解了 ${\ m E}_8$ 型李代数的两个最大刚性轨道，并计算了特征 $p>5$ 的代数闭域上对应约化包络代数中小模的数量。
Multiplicity-free primitive ideals associated with rigid nilpotent orbits
证明了任意有限W-代数U（g，e）都有一个一维表示，它由e的中心化子的分量群的作用固定。因此，对于g中的任何幂零轨道O，存在泛包络代数U（g）的一个无重数（因此是完全素的）本原理想，其相关簇与O的Zebriki闭包一致。
Irreducible representations of finitely generated nilpotent groups
证明了Parshin所证明的群生成的幂零群群的不可约复表示是单性的当且仅当它们有有限权.注意，我们考虑（可能是无限维的）没有任何拓扑结构的表示。此外，我们还证明了对于某些诱导表示，Schur不可约性蕴含不可约性.这两个结果都是在一个更一般的形式表示在任意领域。
In the paper we study finitely generated linear groups of finite rank which have faithful irreducible primitive representations over a field of characteristic zero. We prove that if an infinite finitely generated linear group $G$ of finite rank has a faithful irreducible primitive representation over a field of characteristic zero then the $FC$-center $Δ(G)$ of $G$ is infinite.
Irreducible representations of certain nilpotent groups of finite rank
本文研究了有限阿贝尔全秩幂零群的不可约表示，证明了幂零类为2的无挠极小群在特定条件下的忠实不可约表示可由其子群的不可约本原表示诱导得出。
Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S…
We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating $\mathfrak{p}$-adic integrals associated to certain rank varieties of matrices of linear forms.
Representations of the quantum torus and applications to finitely presented groups
本文证明了量子环面（一个场与一个以场为中心的自由阿尔贝群的交叉积）上强完整模的一个结构定理。这一结果可用于给出n-表现交换幂零群的一个结构定理。
Nilpotent representations of classical quantum groups at roots of unity
适当地专门化“Schnizer模块”中的参数，对于类型A，B，C和D，我们得到其唯一的原始向量。证明了由原向量生成的模是有限维经典量子群在单位根上的不可约最高权模。
We prove a characterization of monomial projective representations of finitely generated nilpotent groups. We also characterize polycyclic groups whose projective representations are finite dimensional.
Primitive ideals, non-restricted representations and finite W-algebras
证明了复半单李代数g中与幂零元e相关的有限W-代数都有有限维表示.为了得到这个结果，我们建立了U（g）的本原理想与有限特征的代数闭域上与e相关的既约包络代数的有限维表示之间的联系.
Enhanced Nilpotent Representations of a Cyclic Quiver
我们定义了一组“增强”幂零锥表示，推广了增强幂零锥。这个集合允许一个由一个具有多个轨道的关联代数群$K$作用。我们定义了一个组合集，参数化的轨道集下，这一行动，我们得出一个纯粹的组合公式的轨道的尺寸。
Let p be a prime, K a p-adic field, G a nilpotent, uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by the centraliser of the second term in the upper central series for G.
Primitive elements in the Hall algebra of a cyclic quiver
本文研究循环箭图幂零表示的Hall代数，给出了其中本原元素的显式公式，解决了该类代数中本原元的具体构造问题。
On some representations of nilpotent Lie algebras and superalgebras
设$G$是一个单连通幂零李群，其李代数为$\gee$ .群$G$通过余伴随作用作用于对偶空间$\gee^*$。%%将$\gee^*$划分为余伴轨道。利用Kirillov的轨道方法，$G$的简单酉表示与$\gee^*$的余伴随轨道是双射对应的，而余伴随轨道又与$\gee$的泛包络代数的本原理想是双射对应的。对于$\gee$的特定子代数具有公共本征向量并且被特定本原理想$I$湮没的简单$\gee$ -模的数目由Benoist证明依赖于对应于$I$的余伴随轨道的某个子簇的几何性质。我们确定的确切数目，这样的模块时，共伴随轨道是二维的。Bell和Musson证明了由分次本原理想分解李超代数的泛包络超代数得到的代数同构于Weyl代数和Clifford代数的张量积。我们描述了某些情况下的因素是纯粹的Weyl代数，并确定这些Weyl代数的大小取决于分次本原理想。
A Note On Nilpotent Representations
设$Γ$是一个非生成的幂零群群，G是一个复约化代数群.表示变种$\mathrm{Hom}(Γ,G)$和字符变种$\mathrm{Hom}(Γ,G)//G$每个携带一个自然的拓扑结构，我们描述的拓扑结构的连接组件的表示分解通过$Γ$的子由其下中心序列的元素。
We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.
Nilpotent orbits: finiteness, separability and Howe's conjecture
本文研究局部非阿基米德域上约化群的幂零轨道。在本文中，我们将试图确定哪些群体只有100多个幂零轨道，哪些群体的幂零轨道是可分的，以及哪些群体豪猜想成立。对于一般的约化群，我们得到了一些部分结果。对于分裂约化群，我们根据根数据和基础局部域的特征得到了一个分类。
From primitive spaces of bounded rank matrices to a generalized Gerstenhaber theorem
在对基域的基数作适当的条件下，由有界秩矩阵本原空间上的Atkinson结构定理，导出了Gerstenhaber定理在幂零矩阵空间上的一个新的推广.
This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert $C^*$-module. An important ingredient in the…
Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6. This is a revised version.
For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $Γ$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(Γ,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to the order of the Weyl group $W$. We give explicit formulas for the Poincaré series. In addition we study $G$-equivariant stable decompositions of subspaces $X(q,G)$ of the free monoid $J(G)$ generated by the Lie group $G$, obtained from finitely generated free nilpotent group rep…
Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras
我们考虑与经典李代数中偶重幂零元相关联的有限W-代数U（g，e）。利用有限W-代数的最高权理论，给出了具有整中心特征标的有限维不可约U（g，e）-模的一个分类.作为推论，我们得到了U（g）的本原理想的参数化，并给出了相应的簇，e的伴随轨道的闭包和积分中心特征标.
Mixed Hodge structures on character varieties of nilpotent groups
本文研究了幂零群到约化代数群的表示簇与特征簇，确定了其上的混合霍奇结构，并给出混合霍奇多项式的显式闭式与递归公式。
Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.
Rational group algebras of finite groups: from idempotents to units of integral group rings
给出了有限幂零群的有理群代数的正交本原幂等元完备集的一个无特征标的显式构造，并给出了这类代数的Wedderburn分解的完整描述.一个直接的后果是一个著名的结果Roquette的舒尔指数的简单组件的群代数有限幂零群群。作为应用，我们得到了有限幂零群$G$的整群环$\Z G$的单位群有一个由三个幂零群生成的有限指数子群，并给出了它们的生成元的明确描述.另一个应用是单位群中自由子群的新构造。在所有的结构处理，对子群$(H,K)$，称为强Shoda对，和明确的建设中心元素$e(G,H,K)$起着至关重要的作用。对任意有限群，证明了有理群代数的本原中心幂等元是这样的$e(G,H,K)$的有理线性组合，且$G$的子群中有$(H,K)$强Shoda偶.
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