All Questions
87
questions
5
votes
0
answers
126
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 2,968
8
votes
0
answers
278
views
Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 1,139
2
votes
0
answers
152
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,711
4
votes
0
answers
270
views
GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,711
6
votes
1
answer
239
views
Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
- 13.8k
6
votes
2
answers
362
views
Twisted forms with real points of a real Grassmannian
Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
- 13.8k
1
vote
0
answers
218
views
Reference request: Weil's uniformization theorem
The Weil uniformization theorem says that if $k$ is an algebraically closed field, $G$ a reductive group, and $C$ a curve, we have an isomorphism of stacks $Bun_G(C)\cong G(F_C)\backslash G(\mathbb{A}...
- 1,837
4
votes
0
answers
132
views
Reference request & more: compute vector bundles for homogeneous $G$-varieties
We work over the field of complex numbers $\mathbb C$.
Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. ...
- 381
5
votes
1
answer
290
views
Reductive groups over positive characteristics
Let $G$ be a connected split reductive group over a field $k$ of characteristic $p$. Let $\mathfrak{g}:=T_e(G)$ denote its Lie algebra. Let $T$ be a maximal split torus and $W$ the Weyl group (of the ...
- 2,711
5
votes
0
answers
250
views
Reference/list of reductive subgroups of reductive groups?
Let $G$ be a (say, connected) reductive group over an algebraically closed field of characteristic zero (say, $\mathbb C$).
I am looking for simple examples of (ideally) complete characterizations of ...
- 3,769
1
vote
0
answers
167
views
When the action of reductive group on algebraic variety is not equidimensional?
I saw the question When is an almost geometric quotient flat? which said
"The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth".
I am curious is there an ...
- 27
12
votes
2
answers
1k
views
Reference request: Elementary proof of Lang's theorem
A few months ago, I read a nice elementary proof of Lang's theorem:
Theorem: Let $G$ be a connected linear algebraic group over $\overline{\mathbb{F}}_p$ and let $F : G \to G$ be a Frobenius map. Then ...
- 741
8
votes
0
answers
282
views
Does Borel fixed-point theorem hold for Deligne-Mumford stacks?
Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus.
Question: Is the following statement true?
...
6
votes
2
answers
479
views
Quasisplit but not split semisimple groups
In section 35.1 of the book "Linear algebraic groups" by Humphreys, it is stated that the quasi-split but not split semisimple groups can only arise when the root system admits a nontrivial ...
- 321
5
votes
2
answers
403
views
Invariant theory over $\mathbb R$
$\DeclareMathOperator\SO{SO}$Suppose we have a (continuous) linear action of $\SO(n,\mathbb R)$ on a vector space $\mathbb R^N$. Consider the ring of invariants $A\subset \mathbb R[x_1,\ldots, x_N]$, ...
- 14.1k