All Questions
14 questions with no upvoted or accepted answers
10
votes
0
answers
881
views
Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties
BACKGROUND:
Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting ...
- 52.9k
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,180
8
votes
1
answer
382
views
Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
- 4,902
6
votes
0
answers
3k
views
Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field
Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two.
I am looking for a reference that explains how to ...
5
votes
0
answers
140
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 ...
- 3,230
5
votes
0
answers
263
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 ...
- 3,799
5
votes
0
answers
118
views
Where to read about the toric variety coming from a principal nilpotent element of a (semi)simple algebraic group?
Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then ...
- 18.4k
4
votes
0
answers
306
views
Computing the $2$-adic volume of a special orthogonal group
Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\...
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516
2
votes
0
answers
245
views
Reference request: proofs of the theorems in the paper "On the representation of the group GL(n, K) where K is a local field"
In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
- 6,201
2
votes
0
answers
156
views
Extension of the Hilbert-Mumford Criterion
Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...
- 2,384
1
vote
0
answers
142
views
Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
- 11
1
vote
0
answers
196
views
Reference Help: Matsuki duality Orbits
I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
- 111
0
votes
0
answers
197
views
'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799