Normal forms of nilpotent elements in semisimple Lie algebras

Nilpotent orbits, normality, and Hamiltonian group actions

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The action of $\g'\simeq R[2]$ exponentiates to an action of the corresponding Lie group $G'$ on a $G'$-cover $M'$ of a nilpotent orbit in $\g'$ such that $M$ is open dense in $M'$. We determine all such pairs $(\g\subset\g')$.

八维实空间中 nilpotent 和 semisimple 四向量的分类

Classification of nilpotent and semisimple fourvectors of a real eight-dimensional space

本文研究 SL(8,R) 在 ∧⁴R⁸ 上的作用，分类 nilpotent 和 semisimple 轨道及 Cartan 子空间，semisimple 轨道分为 1441 类，方法基于 Galois 群论。

Nilpotent pairs in semisimple Lie algebras and their characteristics

Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e_1,e_2) consists of two commuting elements in \g that can independently be contracted to the origin. A principal nilpotent pair is a double counterpart of a regular nilpotent element. Consequently, the theory of nilpotent pairs should stand out as double counterpart of the theory of nilpotent orbits. We show that any nilpotent pair has a characteristic (h_1,h_2), whic…

Semisimple cyclic elements in semisimple Lie algebras

This paper is a continuation of the theory of cyclic elements in semisimple Lie algebras, developed by Elashvili, Kac and Vinberg. Its main result is the classification of semisimple cyclic elements in semisimple Lie algebras. The importance of this classification stems from the fact that each such element gives rise to an integrable hierarchy of Hamiltonian PDE of Drinfeld-Sokolov type.

Weighted Vogan diagrams associated to real nilpotent orbits

We associate to each nilpotent orbit of a real semisimple Lie algebra $g_o$ a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of $g_o$ is noticed in some subalgebra of $g_o$. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements.

On the sum of the index of a parabolic subalgebra and of its nilpotent radical

In this short note, we investigate the following question of Panyushev : ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $\mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author, and proved by Millet-Fauquant and Joseph, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $\mathfrak{g}$. This provides new examples of d…

Principal Nilpotent pairs in a semisimple Lie algebra, I

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmon…

Approximation of nilpotent orbits for simple Lie groups

We propose a systematic and topological study of limits $\lim_{ν\to 0^+}G_\mathbb{R}\cdot(νx)$ of continuous families of adjoint orbits for non-compact simple Lie groups. This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of $\mathrm{SL}_n(\mathbb{R})$ and $\mathrm{SU}(p,q)$ are computed in detail.

Jordan-Chevalley decomposition in Lie algebras

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.

On the existence of homomorphisms between principal series of complex semisimple Lie groups

We determine when there exists a nonzero homomorphism between principal series representations of a complex semisimple Lie group. We also determines the existence of homomorphisms between twisted Verma modules.

Finite convergent presentations of plactic monoids for semisimple lie algebras

We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties.

正则极点半单 Lie 代数

Regular extreme semisimple Lie algebras

本文证明了所有单 Lie 代数均为正则极点，且无非单的半单 Lie 代数是正则极点，推广了 $A_n$ 型的结果。

Cohomology in nonunitary representations of semisimple Lie groups (the group $U(2,2)$)

We suggest a method of constructing special nonunitary representations of semisimple Lie groups using representations of Iwasawa subgroups. As a typical example, we study the group $U(2,2)$.

Small semisimple subalgebras of semisimple Lie algebras

The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak k, \frak g)$, such that for every irreducible finite dimensional $\frak g$-module $V$, there exists an injection of $\frak k$-modules $W \to V$, where $W$ is an irreducible $\frak k$-module of dimension less than $b(\frak k, \frak g)$. This result was announced in math.RT/0310140.

Glider representations of chains of semisimple Lie algebras

We start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring $FR$ and here we consider the right bounded algebra filtration $FU(\mathfrak{g})$ on the universal enveloping algebra $U(\mathfrak{g})$ of some semisimple Lie algebra $\mathfrak{g}$ given by a fixed chain of semisimple sub Lie algebras $\mathfrak{g}_1 \subset \mathfrak{g}_2 \subset \ldots \subset \mathfrak{g}_n = \mathfrak{g}$. Inspired by the classical representation theory, we introduce so-called Verma glider representations. Their exist…

Locally semisimple and maximal subalgebras of the finitary Lie algebras $gl(\infty)$, $sl(\infty)$, $so(\infty)$, and $sp(\infty)$

We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65.

复半单李代数上的对称双导子

Symmetric Biderivations on Complex Semisimple Lie Algebras

本文证明了有限维复半单李代数到其有限维模上的对称双导子均为平凡的，并据此分类了 Takiff 代数、辛振子代数及若干经典单李超代数上的双导子。

Affine slice for the coadjoint action of a class of biparabolic subalgebras of a semisimple Lie algebra

In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.

ad-Nilpotent positively-graded Borel module subalgebras

In this paper, we study certain ad-nilpotent subalgebras contained in the non-zero graded portion of a simple Z_n-graded Lie algebra. These subalgebras respect the grading on the Lie algebra and are modules for a Borel subalgebra for the grade-zero Lie subalgebra. We show that semisimple elements in such subalgebras lie in the center of the subalgebra, and we provide a classification of these subalgebras whose weight space decompositions have only non-zero weights.

Differential equations and the algebra of confluent spherical functions on semisimple Lie groups

We consider the notion of a confluent spherical function on a connected semisimple Lie group, $G,$ with finite center and of real rank $1,$ and discuss the properties and relationship of its algebra with the well-known Schwartz algebra of spherical functions on $G.$

An explicit rational structure for real semisimple Lie algebras

We construct a convenient basis for all real semisimple Lie algebras by means of an adapted Chevalley basis of the complexification. It determines rational and in fact half-integer structure constants which we express only in terms of the root system and the involution that defines the real structure.

A Cohomological Proof that Real Representations of Semisimple Lie Algebras Have $\mathbb{Q}$-Forms

A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.

On Kirillov's lemma for nilpotent Lie algebras

We establish a sharpening of Kirillov's lemma on nilpotent Lie algebras with 1-dimensional center and use it to study the structure of 3-step nilpotent Lie algebras.

Multiplicity free gradings on semisimple Lie and Jordan algebras and skew root

A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are iden…

An Erdos-Turan Inequality For Compact Simply-Connected Semisimple Lie Groups

The classical Erd{\" o}s-Turan Inequality bounds how far a sequence of points in the circle is from being equidistributed in terms of its exponential moments. We prove an analogous inequality for all compact simply-connected semisimple Lie groups, bounding how far a sequence is from being equidistributed in the conjugacy classes of the group in terms of the moments of irreducible characters.

Real representations of semisimple Lie algebras have Q-forms

We prove that each real semisimple Lie algebra G has a Q-form, such that every real representation of G can be realized over the rational numbers Q. This was previously proved by M.S.Raghunathan (and rediscovered by P.Eberlein) in the special case where G is compact.

Series Analysis and Schwartz Algebras of Spherical Convolutions on Semisimple Lie Groups

We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(λ),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^{p}-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^{p}-$ Schwartz functions.

The local character expansion near a tame, semisimple element

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected nor that the underlying field has characteristic zero.

The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser

In this paper, we study the index for several natural classes of non-reductive subalgebras of semisimple Lie algebras. Namely, we look at parabolic subalgebras, centralisers of nilpotent elements, and the normalisers of the centralisers. We discuss a conjecture of Elashvili to the effect that the index of any centraliser is equal to the rank of the semisimple algebra in question. It is shown that Elashvili's conjecture is true for `small' and `large' orbits. Some properties of the index for the normaliser of the centraliser are proved. In particular, we prove that, for a regular nilpotent elem…

Induced nilpotent orbits of the simple Lie algebras of exceptional type

We describe algorithms for computing the induced nilpotent orbits in semisimple Lie algebras. We use them to obtain the induction tables for the Lie algebras of exceptional type. This also yields the classification of the rigid nilpotent orbits in those Lie algebras.

A characterization of nilpotent orbit closures among symplectic singularities

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the Kirillov-Kostant form.

Orbits in real $\Z_m$-graded semisimple Lie algebras

In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this method.

Nilpotent orbits and finite W-algebras

In recent years, the finite W-algebras associated to a semisimple Lie algebra and its nilpotent element have been studied intensively from different viewpoints. In this lecture series, we shall present some basic constructions, connections, and applications of finite W-algebras.

Harish-Chandra highest weight representations of semisimple Lie algebras and Lie groups

In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.

On related varieties to the commuting variety of a semisimple Lie algebra

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.

Semisimple Lie groups satisfy property RD, a short proof

We give a short elementary proof of the fact that connected semisimple real Lie groups satisfy property RD. The proof is based on a process of linearization.

On the semisimple orbits of restricted Cartan type Lie algebras $W, S$ and $H$

In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras $W, S, H$.

Primitive ideals in the affinoid enveloping algebras of semisimple Lie algebras

For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the annihilator of an affinoid highest weight module.

A canonical dimension estimate for non-split semisimple p-adic Lie groups

We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple p-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov-Wadsley and Schmidt in the split semisimple case.

Birational geometry for the covering of a nilpotent orbit closure

A nilpotent orbit $O$ of a complex semisimple Lie algebra $\mathfrak{g}$ has finite fundamental group. Associated with an etale cover of $O$, we have a finite cover of the closure $\bar{O}$ of $O$. In this article we consider the finite cover X associated with the universal cover of a nilpotent orbit $O$ of a classical simple Lie algebra $\mathfrak{g}$. We construct explicitly a Q-factorial terminalization of X in a group theoretic way.

On the nullcone and the variety $χ_\mathfrak{g}$ of a semisimple lie algebra

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x, y) in g\timesg such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and irreducible subvariety of g \times g whose normalization has rational singularities and such that the normalization morphism is bijective.

Restrictions of irreducible characters of finite groups of Lie type to derived subgroups and regular semisimple elements

In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.

Very nilpotent basis and n-tuples in Borel subalgebras

A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.

Special representations of nilpotent Lie groups and the associated Poisson representations of current groups

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extension…

Regular functions on spherical nilpotent orbits in complex symmetric pairs: classical non-Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G, K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality and we describe the K-module structure of the ring of regular functions of the normalizations.

Sharp Fourier type and cotype with respect to compact semisimple Lie groups

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

Faithful representations of infinite-dimensional nilpotent Lie algebras

For locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach-Lie algebras we get norm continuous representations by bounded operators on Banach spaces.

Geometry of central extensions of nilpotent Lie algebras

We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study of the orbit space geometry of the automorphism group action on Grassmannians defined in terms of the second cohomology of the extendable nilpotent Lie algebra. Such a geometric method allows to classify some classes of nilpotent Lie algebras. The concept of a rigid central extension is introduced.

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra g, and let K_C --> Aut(p_C) be the complexified isotropy representation at the identity coset of the corresponding symmetric space G/K. Suppose that O is a nilpotent G-orbit in g and O* is the nilpotent K_C-orbit in p_C associated to O by the Kostant-Sekiguchi correspondence. We show that the complexity of O* as a K_C variety measures the failure of the Poisson algebra of smooth K-invariant functions on O to be commutative.

Loop Grassmannian cohomology, the principal nilpotent and Kostant theorem

Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last sectio…

具有半单包络李代数的Lie-Yamaguti代数的表示

Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

本文研究了包络李代数为半单的Lie-Yamaguti代数的表示问题，利用其包络李代数的表示及附加数据刻画了原代数的表示，并推广至无穷小$s$-流形情形。

幂零李代数的指数

The index of nilpotent Lie algebras

本文研究多类幂零李代数的指数，并针对自由幂零、图及丝状幂零李代数给出显式计算公式。

On integrability of transverse Lie-Poisson structures at nilpotent elements

We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform construction of these integrable systems for an infinite family of distinguished nilpotent elements of semisimple type.

The centralizer of a nilpotent section

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is *very good*. If X in Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/C^o both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type. The proofs are achieved by studying the centralizer C of a nilpotent s…