Hermitian Characteristics of Nilpotent Elements

Fully commutative elements and spherical nilpotent orbits

Let g be a simple Lie algebra, with fixed Borel subalgebra b and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their connections with the spherical nilpotent orbits in g. If g is not of type G_2, it is shown that an element w in W is fully commutative if and only if the subalgebra of b determined by the inversions of w lies in the closure of a spherical nilpotent orbit. A similar characterization is also given for the ad-nilpotent ideals of b, which are parametrized by suitable elem…

Nilpotent Elements of Vertex Algebras

Using the method of commutative algebra, we show that the set $\mathfrak{R}$ of nilpotent elements of a vertex algebra $V$ forms an ideal, and $V/\mathfrak{R}$ has no nonzero nilpotent elements.

Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.

Composantes irréductibles de la variété commutante nilpotente d'une algèbre de Lie symétrique semi-simple

Let θbe an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,θ) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p-distinguished elements. This conjecture was established by A. Premet in the case (g \times g, θ) where θ(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases.

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…

Local Systems on Classical Nilpotent Orbits and Maximal Length Elements

It is known that there is a bijection between dominant weights of a complex reductive Lie group $G$ and the set $\mathcal{N}_{\mathcal{O},r}$ whose elements are of the form $(\mathcal{O},ρ)$, where $\mathcal{O}$ is a nilpotent orbit and $ρ$ is an irreducible, algebraic representation of the stabilizer group $G^e$ of an element $e$ in the nilpotent orbit $\mathcal{O}$. We would like to study the above bijection when $G$ is classical and $ρ$ corresponds to a local system of $\mathcal{O}$. In particular, we will prove Conjecture 3.1 in \cite{AS} and Conjecture 7.4' in \cite{Ac2} in the classical …

The centralisers of nilpotent elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_α$ of elements $α\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $x\in so_8$ such that the co…

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.

奇幂零元与$\mathfrak{gl}(m|n)$中的$\mathfrak{osp}(1|2)$-子代数

Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$

本文研究$\mathfrak{gl}(m|n)$中奇幂零元嵌入$\mathfrak{osp}(1|2)$-子代数的条件，通过定义超Jordan矩阵并证明充要条件：该元位于仅含奇数阶超Jordan块的超Jordan矩阵轨道中。

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.

埃尔米特李群的海森堡抛物诱导表示，第二部分：次极小表示与分支规则

Heisenberg parabolically induced representations of Hermitian Lie groups, Part II: Next-to-minimal representations and branching rules

本文证明由海森堡型极大抛物子群诱导的球面表示族端点处的可酉化子表示是次极小表示，并研究了其在SO₀(2,n)和E₆₍₋₁₄₎中的分支规则。

Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.

Normalized intertwining operators and nilpotent elements in the Langlands dual group

Let $F$ be a local non-archimedian field and let $G$ be a group of points of a split reductive group over $F$. For a parabolic subgroup $P$ of $G$ we set $X_P=G/[P,P]$. For any two parabolics $P$ and $Q$ with the same Levi component $M$ we construct an explicit unitary isomorphism $L^2(X_P)\to L^2(X_Q)$ (which depends on a choice of an additive character of $F$). The formula for the above isomorphism involves the action of the principal nilpotent element in the Langlands dual group of $M$ on the unipotent radicals of the corresponding dual parabolics. We use the above isomorphisms to define a …

Parabolic orbits of $2$-nilpotent elements for classical groups

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a representation-theoretic context in the language of a symmetric quiver algebra. This makes it possible to provide a parametrization of the orbits via a combinatorial tool that we call symplectic/orthogonal oriented link patterns. We deduce information about numerology. We then generalize these classifications to standard parabolic subgroups for all classical groups. Finally, our r…

Centers of centralizers of nilpotent elements in exceptional Lie superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;α)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfra…

On Dynkin gradings in simple Lie algebras

In this paper we study gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. We show that for each odd nilpotent element there always exists canonically associated "strictly" odd nilpotent element, which allows us to reduce our investigations to the latter.

On cocharacters associated to nilpotent elements of reductive groups

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J.C. Jantzen.

Representatives for unipotent classes and nilpotent orbits

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…

Abundance of nilpotent orbits in real semisimple Lie algebras

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace of the longest Weyl group element. The result is used to the study of fundamental groups of non-Riemannian locally symmetric spaces.

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $σ$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $σ$-minuscule elements in the affine Weyl …

Existence of affine pavings for varieties of partial flags associated to nilpotent elements

The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…

Centralizers of nilpotent elements in basic classical Lie superalgebras in good characteristic

Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the reductive algebraic group over \mathbb{K} such that \mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}) by …

Centres of centralizers of nilpotent elements in Lie superalgebras $\mathfrak{sl}(m|n)$ or $\mathfrak{osp}(m|2n)$

Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or $\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let $\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a superalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular, we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and $\left(\mathfrak{z}(\ma…

Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are…

二阶幂零元中心化子在旗簇中的轨道闭包：A、B、D 型的正规性与消解

Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

本文研究 A、B、D 型约化群中二阶幂零元中心化子在旗簇内的轨道闭包，证明其正规性、特定条件下的 Cohen–Macaulay 性与有理奇点，并构造涉及 Schubert 簇的双有理有理消解。

论 Borel 子代数的幂零理想中的伴随轨道

On adjoint orbits in nilpotent ideals of a Borel subalgebra

本文研究复半单 Lie 代数 Borel 子代数幂零理想中的伴随轨道，证明由确定该最小理想的幂零元组成的集合恰有一个闭 Borel 子群轨道，并用相关极小根刻画其代表元。

Ideals of Heisenberg type and minimax elements of affine Weyl groups

We consider ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra. The goal of this paper is two-fold. First, we study the ad-nilpotent ideals lying inside the Heisenberg ideal. The Heisenberg ideal is the nilpotent radical of the centralizer of the highest root vector. Second, we study the ideals having the property that the corresponding domain of the Shi arrangement consists of a single alcove. Such ideals (and the corresponding elements of the affine Weyl group) are called minimax.

On divisible weighted Dynkin diagrams and reachable elements

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra $\g$. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in $\g$ determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkabl…

Indice du normalisateur du centralisateur d'un element nilpotent dans une algebre de Lie semi-simple

The index of a complex Lie algebra is the minimal codimension of its coadjoint orbits. Let us suppose $\g$ semisimple, then its index, ${\rm ind} \g$, is equal to its rank, ${\rm rk \g}$. The goal of this paper is to establish a simple general formula for the index of $\n(\g^ξ)$, for $ξ$ nilpotent, where $\n(\g^ξ)$ is the normaliser in $\g$ of the centraliser $\g^ξ$ of $ξ$. More precisely, we have to show the following result, conjectured by D. Panyushev \cite{Panyushev} : $${\rm ind} \n(\g^ξ) = {\rm rk \g}-\dim \z(\g^ξ),$$ where $\z(\g^ξ)$ is the center of $\g^ξ$. D. Panyushev obtained in \ci…

Levi decomposition of nilpotent centralisers in classical groups

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].

Ad-nilpotent ideals of a Borel subalgebra: generators and duality

It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral points of a simplex with rational vertices. We give a description of the generators of ad-nilpotent ideals in terms of these elements, and show that an ideal has $k$ generators if and only it lies on the face of this simplex of codimension $k$. We also consider two combinatorial statistics on the set of ad-nilpotent ideals: the number of simple roots in the ideal and the number of generators. Considering …

K-orbits on the flag variety and strongly regular nilpotent matrices

In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i…

On the centralizer of the sum of commuting nilpotent elements

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F=K(t) is the field of rational functions on the projective line, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a:b) of a suitable open subset of the projective line, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX+bY. This answers a question of J. Pevtsova.

Ad-Nilpotent Ideals of a Parabolic Subalgebra

We extend the results of Cellini-Papi on the characterizations of nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical Lie algebra, we give formulas for the number of these ideals.

Adapted pairs in type $A$ and regular nilpotent elements

Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of semi-invariants on $\mathfrak q^*$ is polynomial in most cases, in particular when $\mathfrak g$ is simple of type $A$ or $C$. On the other hand $\mathfrak q$ admits a canonical truncation $\mathfrak q_Λ$ such that $Sy(\mathfrak q)=Sy(\mathfrak q_Λ)=Y(\mathfrak q_Λ)$ where $Y(\mathfrak q_Λ)$ denotes the algebra of invariant functions on $\mathfrak q_Λ^*$. An ada…

On exceptional nilpotents in semisimple Lie algebras

We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.

A Note On Nilpotent Representations

Let $Γ$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(Γ,G)$ and the character variety $\mathrm{Hom}(Γ,G)//G$ each carry a natural topology, and we describe the topology of their connected components in terms of representations factoring through quotients of $Γ$ by elements of its lower central series.

Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups

For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal-Bargmann tran…

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter n…