Related papers: Computing with nilpotent orbits in simple Lie alge…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the…

Let $G$ be a connected reductive group acting on a complex vector space $V$ and projective space ${\mathbb P}V$. Let $x\in V$ and ${\cal H}\subseteq {\cal G}$ be the Lie algebra of its stabilizer. Our objective is to understand points…

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…

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…

We classify the nilpotent Lie algebras of real dimension eight and minimal center that admit a complex structure. Furthermore, for every such nilpotent Lie algebra $\mathfrak{g}$, we describe the space of complex structures on…

A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four…

In this paper the authors introduce an analog of the nilpotent cone, ${\mathcal N}$, for a classical Lie superalgebra, ${\mathfrak g}$, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical…

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the…

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating…

Let $\g$ be simple Lie algebra. We give a conceptual proof for the fact that the nilpotent orbits of height 3 are spherical. It is shown that if the highest root of $\g$ is fundamental, then $\g$ has a specific nilpotent orbit of height 3.…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on…

This paper includes the classification, in a simple Lie algebra, of the singularities of Slodowy slices between special nilpotent orbits that are adjacent in the partial order on nilpotent orbits. The irreducible components of most…

Kirillov's orbit theory provides a powerful tool for the investigation of irreducible unitary representations of many classes of Lie groups. In a previous paper we used a modification hereof, called monomial linearisation, to construct a…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra.…

We extend results on finite dimensional nilpotent Lie algebras to Leibniz algebras and counterexamples to others are found. One generator algebras are used in these examples and are investigated further.

In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for…

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.