Related papers: Richardson elements for parabolic subgroups of cla…

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an…

Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie…

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$…

Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n \in \mathbb Z_{>0}$, and take $G$ to be one of the classical algebraic groups $\mathrm{GL}_n(k)$, $\mathrm{SL}_n(k)$, $\mathrm{Sp}_n(k)$, $\mathrm O_n(k)$ or…

Let $G$ be a connected semisimple algebraic group with Lie algebra $g$ and $P$ a parabolic subgroup of $G$ with $Lie(P)=p$. The parabolic contraction of $g$ is the semi-direct product of $p$ and a $p$-module $g/p$ regarded as an abelian…

Let $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^{\theta})_{0} $ the identity component of the fixed points of an involutive automorphism $ \theta $ of $ G $. The pair $ (G, K) $ is called a symmetric…

We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…

Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable…

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…

Let $\mathbb{K}$ be an algebraically closed field of characteristic 0. A finite dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ is said to be stable if there exists a linear form $g\in\mathfrak{g}^{*}$ and a Zariski open subset in…

We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the…

We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the…

We show that, in a weakly regular $p$-adic Lie group $G$, the subgroup $G_u$ spanned by the one-parameter subgroups of $G$ admits a Levi decomposition. As a consequence, there exists a regular open subgroup of $G$ which contains $G_u$.

An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic…

Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the…

Let $\mathbf{P}$ be a parabolic subgroup with Levi $\mathbf{M}$ of a connected reductive group defined over a locally compact non-archimedean field $F$. Given a certain compact open subgroup $\Gamma$ of $\mathbf{P}(F)$, this note proves…

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…

Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi…

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is…

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call…