All Questions
14 questions
2
votes
0
answers
158
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
8
votes
1
answer
221
views
Coordinates on $N_+ \backslash \overline{B_+ w B_+} / N_+$
Let $G = \text{GL}_n(\mathbb{C})$ and let $N_+$ be the subgroup of upper triangular matrices with $1$'s on the diagonal. Let $w$ be a permutation, let $B_+ w B_+$ be the Bruhat cell and let $\overline{...
- 156k
1
vote
0
answers
107
views
Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure
In this question, the following fact was used by the respondent
A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar
measure contains a coset of $G^0$, the connected component of
$G$ ...
- 63.9k
12
votes
1
answer
684
views
Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?
Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
I was told some fact along this line is true but could not find any ...
- 31.5k
8
votes
3
answers
2k
views
Cohomology rings of $ GL_n(C)$, $SL_n(C)$
Can anyone provide me with the reference for the following fact
(idea of the proof will be appreciated too):
Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
- 774
7
votes
1
answer
387
views
Chevalley restriction theorem for non-split Cartan
Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...
- 15.5k
6
votes
1
answer
456
views
How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?
I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
- 11.2k
3
votes
2
answers
285
views
Invariant theory for parabolics
Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...
- 8,597
4
votes
1
answer
677
views
Lifting one parameter subgroups of algebraic groups
Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...
- 1,389
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
0
votes
0
answers
99
views
Unions of orbits of dimension $\leq n$
Let $G$ be a complex linear algebraic group acting on a smooth complex projective variety $X$ with finitely many orbits. Note that each $G$-orbit is a smooth locally closed subvariety of $X$.
For a ...
- 4,920
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 ...
- 2,639
1
vote
1
answer
308
views
Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)
I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: U^...
CommunityBot
- 1
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 11.2k