Related papers: Equations for some nilpotent varieties

In Lie theory, a dense orbit in the unipotent radical of a parabolic group under the adjoint action is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalising the classical definition in the case of…

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…

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 $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy…

We construct an L^2-model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If $V$ is split and G is not of type A_n, then…

Let $G$ be a quasi-simple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint representation. It is known…

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q} under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter number of…

For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In…

In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it…

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for…

In this paper, we investigate the conditions under which an odd nilpotent element in $\mathfrak{gl}(m|n)$ lies inside an $\mathfrak{osp}(1|2)$-subalgebra. In the case of the classical Lie algebra $\mathfrak{gl}_m$, every nilpotent element…

Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

Let $G$ be a simple algebraic group and $\ggg=\Lie(G)$ over $k=\bar\bbf_q$ where $q$ is a power of the prime characteristic of $k$, and $F$ a Frobenius morphism on $G$ which can be defined naturally on $\ggg$. In this paper, we investigate…

Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive…

Let $\mathfrak{g}$ be a simple classical Lie algebra over $\mathbb{C}$ and $G$ be the adjoint group. Consider a nilpotent element $e\in \mathfrak{g}$, and the adjoint orbit $\mathbb{O}=Ge$. The formal slices to the codimension $2$ orbits in…

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…

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary…

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable,…

We introduce the notion of Groebner S-basis of an ideal of the free associative algebra K<X> over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Groebner S-bases of the ideal…

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…