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Questions tagged [linear-algebraic-groups]

The tag has no usage guidance, but it has a tag wiki.

7
votes
1answer
232 views

“Almost-ideals” in the (simple) Lie algebra of an algebraic group?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple. Is it necessarily the case that ...
3
votes
1answer
201 views

Definition of simple linear algebraic group

Why is it that many sources define simple (or almost-simple) linear algebraic group $G/k$ to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is ...
4
votes
2answers
225 views

How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...
7
votes
1answer
167 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
93 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
98 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
116 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
118 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
172 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 $G$-...