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7
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1answer
156 views

Simple Lie algebras: making subspaces 'very transversal'

Let $G$ be a Lie group or group of Lie type whose Lie algebra $\mathfrak{g}$ is simple. Because the Lie algebra is simple, for any proper subspace $V\subset \mathfrak{g}$, there is a $g\in G$ such ...
5
votes
0answers
88 views

$\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type

I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...
1
vote
1answer
91 views

Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

Let $\newcommand{\GG}{\mathbf{G}}\newcommand{\g}{\mathfrak{g}}\GG$ be a connected semisimple algebraic group over the algebraically closed field $k=\overline{\mathbb{F}_q}$, and let $\g$ be its Lie-...
3
votes
1answer
115 views

Is the subgroup generated by a conjugacy class of semisimple elements Zariski closed?

Let $k$ be an arbitrary field with $\operatorname{char}(k) \neq 2$. Let $G$ be a linear algebraic group over $k$. Let $X$ be the conjugacy class of a semisimple element $s \in G(k)$ of order 2 (or a ...
1
vote
0answers
106 views

Integral smooth model of unramified reductive groups

My question is motivated by the following observations. Let $\mathrm{T}$ be a torus defined over a $p$-adic field $K$, then by theories of tori, we have it is uniquely determined by a free $\mathbb{Z}$...
4
votes
0answers
167 views

A basic question on a base change of a homogeneous space of a linear algebraic group

I asked this basic question in MSE and got a comment "This belongs to Mathoverflow", so I ask my question here. Let $G$ be a linear algebraic group over a field $k$, and $H\subset G$ be a $k$-...
0
votes
1answer
69 views

Haar measure of algebraic orbits

Let $G$ be a simple linear algebraic group acting on a projective variety $X$ through rational maps. Let $x_0\in X$ with stabilizer group $H$ and assume that $G/H$ in not compact and carries a $G$-...