Dimension of a minimal nilpotent orbit

B-orbits of 2-nilpotent matrices and generalizations

The orbits of the group B of upper-triangular matrices acting on 2-nilpotent complex matrices via conjugation are classified via oriented link patterns, generalizing A. Melnikov's classification of the B-orbits on upper-triangular such matrices. The orbit closures as well as the "building blocks" of minimal degenerations of orbits are described. The classification uses the theory of representations of finite-dimensional algebras. Furthermore, we initiate the study of the B-orbits on arbitrary nilpotent matrices.

Unipotent Representations Attached to the Principal Nilpotent Orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations.

Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

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.

Simple modules for Affine vertex algebras in the minimal nilpotent orbit

We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.

Nilpotent orbits in classical Lie algebras over $\textbf{F}_{2^n}$ and the Springer correspondence

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of $G$.

Universal formula for Hilbert series of minimal nilpotent orbits

We show that the Hilbert series of the projective variety $X=P(O_{min}),$ corresponding to the minimal nilpotent orbit $\mathcal O_{min},$ is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters $α,β,γ$ and represents a special case of the generalized hypergeometric function ${}_{4}F_{3}.$ A universal formula for the degree of $X$ is then deduced.

Two-row nilpotent orbits of cyclic quivers

We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free.

On the maximum nilpotent orbit intersecting a centralizer in M(n,K)

To any pair of commuting n x n nilpotent matrices it is associated a pair of partitions of n. We describe a maximal nilpotent subalgebra of the centralizer of a given nilpotent n x n matrix and prove a conjecture of Polona Oblak which consists in an algorithm for the determination of the maximum partition which forms with a given partition a pair with the previous property.

Visible actions on spherical nilpotent orbits in complex simple Lie algebras

This paper studies nilpotent orbits in complex simple Lie algebras from the viewpoint of strongly visible actions in the sense of T. Kobayashi. We prove that the action of a maximal compact group consisting of inner automorphisms on a nilpotent orbit is strongly visible if and only if it is spherical, namely, admitting an open orbit of a Borel subgroup. Further, we find a concrete description of a slice in the strongly visible action. As a corollary, we clarify a relationship among different notions of complex nilpotent orbits: actions of Borel subgroups (sphericity); multiplicity-free represe…

Nilpotent coadjoint orbits in small characteristic

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra $G_2$ in characteristic $3$ and in the dual of exceptional Lie algebra $F_4$ in characteristic $2$ are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type $B,C,F_4$ in characteristic $2$ and in the dual of Lie algebra of type $G_2$ in characteristic $3$. In each case we give an explicit description of the nilpotent pieces in the dual defined in \cite{CP}, which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case…

On Quantization of a Nilpotent Orbit Closure in $G_2$

Let $G$ be the complex exceptional Lie group of type $G_2$. Among the five nilpotent orbits in its Lie algebra $\mathfrak{g}$, only the 8-dimensional orbit $\mathcal{O}_8$ has non-normal orbit closure $\bar{\mathcal{O}_8}$. In this short note, we will give a quantization model of $\bar{\mathcal{O}_8}$, verifying a conjecture of Vogan in 1984.

Admissible modules and normality of classical nilpotent orbits II

In this paper, we compute the character formula of the Brylinski model for all classical nilpotent varieties $\overline{\mathcal{O}}$. As a consequence, one can compute the multiplicities of all $K-$types of the ring of regular functions $R(\overline{\mathcal{O}})$ for all classical nilpotent varieties.

Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence

Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.

Fourier Transforms of Nilpotent, Coadjoint Orbits for GL(n,R)

The main result of this paper is an explicit formula for the Fourier transform of the canonical measure on a nilpotent coadjoint orbit for GL(n,R). This paper also includes some results on limit formulas for reductive Lie groups including new proofs of classical limit formulas of Rao and Harish-Chandra.

Regular Functions of Symplectic Spherical Nilpotent Orbits and their Quantizations

We study the ring of regular functions of classical spherical orbits $R(\mathcal{O})$ for $G = Sp(2n,\mathbb{C})$. In particular, treating $G$ as a real Lie group with maximal compact subgroup $K$, we focus on a quantization model of $\mathcal{O}$ when $\mathcal{O}$ is the nilpotent orbit $(2^{2p}1^{2q})$. With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers for such orbits. Assuming the results in [Barbasch 2008], we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.

经典李代数的 nilpotent 轨道在 negation 下的稳定性

Nilpotent orbits of classical Lie algebras stable under negation

本文研究经典李代数中 nilpotent 轨道在 negation 下的稳定性，证明在这些轨道上不存在 Gibbs 状态，并对所有经典李代数进行了分类。

Nilpotent subspaces and nilpotent orbits

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a disti…

Orbites Nilpotentes Sphériques et Représentations unipotentes associées : Le cas $SL_n$

Let $G$ be a real simple Lie group, $\got g$ its Lie algebra. Given a nilpotent adjoint $G$-orbit $O$, the question is to determine the irreducible unitary representations of $G$ that we can associate to $O$, according to the orbit method. P.Torasso, in [22], gave a method to solve this problem if $O$ is minimal. In this paper, we study the case where $O$ is any spherical nilpotent orbit of $sl_n({\mathbb R})$, we construct, from $O$, a family of representations of the two-sheeted covering of $SL_n({\mathbb R})$ with Torasso's method and, finally, we show that all these representations are ass…

Local Systems on Nilpotent Orbits and Weighted Dynkin Diagrams

We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.

Enhanced Nilpotent Representations of a Cyclic Quiver

We define a set of "enhanced" nilpotent quiver representations that generalizes the enhanced nilpotent cone. This set admits an action by an associated algebraic group $K$ with finitely many orbits. We define a combinatorial set that parametrizes the set of orbits under this action and we derive a purely combinatorial formula for the dimension of an orbit.

与刚性幂零轨道关联的无重数本原理想的数量

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。

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 $(\g\subset\g')$.

Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes.

Corrigendum to `Orbit closures in the enhanced nilpotent cone', published in Adv. Math. 219 (2008)

In this note, we point out an error in the proof of Theorem 4.7 of [P. Achar and A.~Henderson, `Orbit closures in the enhanced nilpotent cone', Adv. Math. 219 (2008), 27-62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.

Quantization of Special Symplectic Nilpotent Orbits and Normality of their Closures

We study the regular function ring $R(\mathcal{O})$ for all symplectic nilpotent orbits $\mathcal{O}$ with even column sizes. We begin by recalling the quantization model for all such orbits by Barbasch using unipotent representations. With this model, we express the multiplicities of fundamental representations appearing in $R(\mathcal{O})$ by a parabolically induced module. Finally, we will use this formula to give a criterion on the normality of the Zariski closure $\bar{\mathcal{O}}$ of $\mathcal{O}$.

On Rational Nilpotent Orbits of $SL_{n}$ and $Sp_{2n}$ over a Local Non-Archimedean Field

We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the split classical groups $SL_n$ and $Sp_{2n}$ over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.

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 …

Springer fibers for the minimal and the minimal special nilpotent orbits

We describe Springer fibers corresponding to the minimal and minimal special nilpotent orbits of simple Lie algebras. As a result, we give an answer to the conjecture of Humphreys regarding some graphs attached to Springer fibers.

Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.

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。

Tempered Representations and Nilpotent Orbits

Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.

sl(n,R)的幂零轨道不承认吉布斯态

Nilpotent orbits of sl(n,R) admit no Gibbs state

本文证明，当n>2时，SL(n,R)的幂零轨道不承认作为哈密顿G-流形上自然平衡态的吉布斯态。

准可接受、可提升的nilpotent轨道和覆盖Barbasch-Vogan对偶性

Quasi-admissible, raisable nilpotent orbits and covering Barbasch-Vogan duality

研究确定了在p-adic局部域F上简单连通李群类型E中，给定F-分裂nilpotent轨道成为准可接受或可提升所需覆盖次数，并证明所有几何类型包含在覆盖Barbasch-Vogan对偶性映射图像中的F-分裂nilpotent轨道总是准可接受。

Hikita conjecture for the minimal nilpotent orbit

We check that the statement of Hikita conjecture holds for the case of the minimal nilpotent orbit of a simple Lie algebra $\mathfrak{g}$ of type ADE and $\mathbb{C} ^2 / Γ$.

Representations associated to small nilpotent orbits for complex Spin groups

This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $ G_ 0 =Spin(2n,\mathbb C)$ be the Spin complex group viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular functions on some small nilpotent orbits $\mathcal O$ transforming according to characters $ψ$ of $C_{ K}(\mathcal O)$ trivial on the connected component of the identity $C_{ K}(\mathcal O)^0$. We then match them w…

关于增强的可约群（II）：在增强群作用下nilpotent轨道的有限性和其闭包

On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

本文研究增强群作用下nilpotent轨道的有限性及闭包几何，通过分类有限nilpotent轨道并构造增强旗流形来描述其闭包。

On the existence of smooth orbital varieties in simple Lie algebras

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers …

A note on étale representations from nilpotent orbits

A linear étale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim V$. A current line of research investigates which étale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of étale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine whi…

Properties of nilpotent orbit complexification

We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra $\mathfrak{g}$ and those in its complexification $\mathfrak{g}_{\mathbb{C}}$. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those $\mathfrak{g}$ having non-empty intersections with all nilpotent orbits in $\mathfrak{g}_{\mathbb{C}}$. Finally, for $\mathfrak{g}$ quasi-split, we characterize those complex nilpotent orbits containing real ones.

On unipotent and nilpotent pieces for classical groups

We show that the definition of unipotent (resp. nilpotent) pieces for classical groups given by Lusztig (resp. Lusztig and the author) coincides with the combinatorial definition using closure relations on unipotent classes (resp. nilpotent orbits). Moreover we give a closed formula for a map from the set of unipotent classes (resp. nilpotent orbits) in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent (resp. nilpotent) pieces.

An orbit model for the spectra of nilpotent Gelfand pairs

Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $Δ(K,N)$ of bounded $K$-spherical functions on $N$ an…

Representations associated to small nilpotent orbits for real Spin groups

The results in this paper provide a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let $\widetilde{G_0} =\widetilde{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\widetilde{K}=Spin(a, \mathbb C)\times Spin(b, \mathbb C)$ be the complexification of the maximal compact subgroup of $\widetilde{G_0}$. We consider the nilpotent orbit $\mathcal O_c$ parametrized by $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ with $k>0$. We provide a list of unipotent representations that are genuine, and prove that t…

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.

Birational geometry for nilpotent orbits

This is a survey article prepared for the submission to "Handbook of moduli". The following topics are discussed: (i) Basic facts and examples of resolutions for nilpotent orbit (ii) Q-factorial terminalizations of nilpotent orbit closures and related birational geometry (iii) Poisson deformations of nilpotent orbit closures.

Regular functions on covers of nilpotent coadjoint orbits

In this paper I explore the relationship between regular functions associated to local systems on nilpotent orbits and unipotent representations in the complex groups.

Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits

Let $\FRAK{g}$ be a classical simple Lie superalgebra. To every nilpotent orbit $\cal O$ in $\FRAK{g}_0$ we associate a Clifford algebra over the field of rational functions on $\cal O$. We find the rank, $k(\cal O)$ of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a $U(\FRAK{g})$-module with $\cal O$ or an orbital subvariety of $\cal O$ as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant $k(\cal O)$ is in many cases, equal to the odd dimension of th…

Jet schemes of the closure of nilpotent orbits

We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Mustautca (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of the…

D 型和 E 型的最小幂零轨道

Minimal Nilpotent Orbits of type D and E

本文证明了 D_n 和 E_6 型最小幂零伴随轨道闭包分别同构于 T^*(SL_{n-1}/[P,P]) 和 T^*(SL_4/P^u) 的仿射化，并为 E_7 型提出类似结果。

Nilpotent cones and adjoint orbits

A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain.

Ad-nilpotent Ideals of Minimal Dimension

n this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and generators, and prove that the equivalence relation is compatible with the left cell structure of affine Weyl group of $\ccg$ and Lusztig's star operator for type $\tilde A_{n-1}$.

Nilpotent orbits: finiteness, separability and Howe's conjecture

This paper is about nilpotent orbits of reductive groups over local non-Archimedean fields. In this paper we will try to identify for which groups there are only finitely many nilpotent orbits, for which groups the nilpotent orbits are separable and for which groups Howe's conjecture holds. For general reductive groups we get some partial results. For split reductive groups we get a classification in terms of the root data and the characteristic of the underlying local field.

On the nilpotent orbits arising from admissible affine vertex algebras

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

B-orbits of nilpotent order 2 and link patterns

In this paper we describe geometry of orbits of upper triangular matrices of nilpotent order 2 under conjugation by the group of upper triangular invertible matrices in terms of link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotent order 2 and intersections of components of a Springer fiber of nilpotent order 2. In particular we connect our results to the combinatorics of meanders and Temperley-Lieb algebras.

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.