Victoria Sevostyanova
In the present paper the field of invariants of the adjoint action of the unitriangular group in the nilradical of any parabolic subalgebra is described.
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In the paper the algebra of invariants of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up a conjecture on the structure of the algebra of invariants. The conjecture is proved for parabolic subalgebras of special types.
In this paper the algebra of invariants for the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We prove that the algebra of invariants is finitely generated.
In the present paper the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up general conjectures on the construction of the field of invariants and the structure of orbits of maximal dimension. The conjecture is proved for parabolic subalgebras of special types.
In this paper the field of invariants for the adjoint action of the Borel group in the nilradical of a parabolic subalgebra is studied. We construct the set of B-invariant rational functions generating the field of invariants.
We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types Bn, Cn, Dn. We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the field of invariants.
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidim…
Saturation rank for nilradical of parabolic subalgebras in Type A
本文通过构造有限连通箭图，组合刻画了A型抛物子代数幂零根中Richardson元的中心化子，并证明在温和条件下幂零根的饱和秩等于SL_{n+1}(K)的半单秩。
B-stable ideals in the nilradical of a Borel subalgebra
本文计算了Borel子代数的诣零根中严格正的$B$ -稳定理想的个数，证明了任何$B$ -稳定理想的极小根都由外尔群的一个元素共轭到单根的子集上.我们还计算了最小根与单根的固定子集共轭的理想的个数。
Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{sl}(V)$ of maximal dimension and let $\mathfrak{n} \subset \mathfrak{p}$ be the corresponding nilradical. In this paper we classify the set of $\mathfrak{sl}(V)$-modules whose restriction to $U(\mathfrak{n})$ is free of rank $1$. It turns out that isomorphism classes of such modules are parametrized by polynomials in $\dim V-1$ variables. We determine the submodule structure for these modules and we show that they generically are simple.
Richardson elements for classical Lie algebras
半单李代数的抛物子代数分解为$\frak{p}=\frak{m}\oplus\frak{n}$，其中$\frak{m}$是Levi因子，$\frak{n}$是相应的零根.根据Richardson定理，在伴随群$P$作用下，存在一个开轨.这个稠密轨道的元素被称为理查森元素。本文描述了经典情形下Richardson元的一个标准形。这将Bruestle，Hille，Ringel和Roehrle关于一般线性群的构造推广到了其它经典李代数，并将作者关于单李代数的优美抛物子代数的Richardson元的正规形推广到了经典李代数的任意抛物子代数.作为应用，我们得到了Richardson元的支集的描述，并得到了Richardson元轨道的Bala-Carter标号。
Let $\mathfrak g$ be a complex simple Lie algebra. We classify the parabolic subalgebras $\mathfrak p$ of $\mathfrak g$ such that the nilradical of $\mathfrak p$ has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of $\mathfrak g$. Some invariant-theoretic consequences of the existence of a commutative polarisation are also discussed.
This paper constructs a novel Hopf algebra $\mathsf{cf}(\mathrm{UT}_{\bullet})$ on the class functions of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of $\mathrm{GL}_{n}(\mathbb{F}_{q})$ class functions. Furthermore, $\mathsf{cf}(\mathrm…
Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
本文研究经典单李代数极大幂零子代数的低维余伴随轨道，以正根子集显式刻画，并据此计算有限域 Chevalley 群极大幂幺子群的指定维数不可约表示数，验证 Isaac 猜想。
Parabolic and Levi subalgebras of finitary Lie algebras
设$\g$是一个局部约化复李代数，它允许一个忠实的可数维有限表示$V$ .这样的李代数是阿贝尔李代数通过$\sl_\infty$、$\so_\infty$、$\sp_\infty$和有限维简单李代数的副本的直和的分裂扩张。$\g$的抛物子代数是包含极大局部可解（即Borel）子代数的任何子代数。在Dimitrov和本文作者工作的基础上，我们用广义旗紧偶的联合稳定子给出了$\g$的抛物子代数的一般描述。与Borel子代数情况的主要区别在于，一般抛物子代数的描述必须同时使用自然模和共自然模，并且抛物子代数在适当的联合稳定子中通过进一步的“迹条件”被挑选出来。紧偶技术也可以用来证明$\gl_\infty$的任意子代数$\k$的Levi分支的存在性。如果$\k$是可裂的，我们证明了线性诣零根在$\k$中存在局部可约补.最后，我们的文件与Cartan，Borel，抛物子代数的任意可裂子代数的$\gl_\infty$的描述。
Let $G$ be a real linear reductive group and $K$ be a maximal compact subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie algebra $\mathfrak{p}$, and $\mathfrak{n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak{n},V_{K})\cong H_{i}(\mathfrak{n},V)$ ($i\geq 0$), where $V_{K}$ denotes the $(\mathfrak{g},K)$ module of $K$ finite vectors in $V$. This is called Casselman's comparison theorem. As a consequence…
The Bruhat order on abelian ideals of Borel subalgebras
本文在特征不是非常坏的代数闭域上，对拟单代数群Borel子代数幂零根中B-稳定阿贝尔子代数内的B-轨道进行参数化，并刻画其闭包关系。
This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass Sections for Parabolic adjoint action in type $A$]. Let $G$ be an irreducible simple algebraic group and $B$ a Borel subgroup of $G$. Let $\mathfrak n$ be the Lie algebra of the nilradical of $B$. Consider an irreducible subgroup $P$ of $G$ containing $B$. Let $P'$ be the derived group of $P$. Let $\mathfrak m$ be the Lie algebra of the nilradical of $P$. A theorem of Richardson asserts that the algebra $\mathbb C[\mathfrak m]^{P'}$ of $P$ semi-invariants is multiplicity-free. A linear subvariety $e+V$ such that the restric…
On the hypersurface orbital varieties of sl(N,C)
我们研究了sl（N，C）的超曲面轨道簇的结构（这些超曲面在一些抛物子代数的零根），以及如何在与之相关的标准Young表中编码这种结构的信息由Robinson-Schensted算法。给出了超曲面轨道簇的唯一非线性定义方程的精确形式的一个猜想，并在一定条件下证明了该猜想。
Let G be a reductive group over an algebraically closed field of positive characteristic. In this article we show an analogue for Morozov theorem for characteristics that are separably good for G (and under additional hypotheses on the group). This theorem characterises, when p= 0, subalgebras of parabolic subgroups of G with respect to their nilradical. If now k is an arbitrary field, let C be a smooth projective and geometrically connected curve. Let G be a reductive C-group scheme, which is the twisted form of a constant C-group. The aforementioned analogue is in particular useful to extend…
We initiate a new line of investigation on branching problems for generalized Verma modules with respect to complex reductive symmetric pairs (g,k). Here we note that Verma modules of g may not contain any simple module when restricted to a reductive subalgebra k in general. In this article, using the geometry of K_C orbits on the generalized flag variety G_C/P_C, we give a necessary and sufficient condition on the triple (g,k, p) such that the restriction X|_k always contains simple k-modules for any g-module $X$ lying in the parabolic BGG category O^p attached to a parabolic subalgebra p of …
The notion of "Weierstrass Section", comes from Weierstrass canonical form for elliptic curves. In celebrated work [B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an "adapted pair" and indeed the construction in [A. Joseph and D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs, Transform. Groups 15 (2010), no. 4, 851-882] works rather well for the coadjoint action of an algebraic, but not…
This work is a continuation of [Fittouhi and Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type $A$]. Let $P$ be a parabolic subgroup of an irreducible simple algebraic group $G$, $P'$ its derived group and $\mathfrak m$ be the nilradical to its Lie algebra. A theorem of Richardson implies that the subalgebra $\mathbb C[\mathfrak m]^{P'}$, spanned by the $P$ semi-invariants in $\mathbb C[\mathfrak m]$, is polynomial. A linear subvariety $e+V$ of $\mathfrak m$ is is called a Weierstrass section for the action of $P'$ on $\mathfrak m$, if the restrictio…
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