All Questions
Tagged with lie-groups algebraic-groups
349 questions
6
votes
0
answers
200
views
Why should Serre's conjecture on congruence subgroup property hold?
There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
Whether a lattice in the group satisfies the congruence subgroup property,
...
- 1,024
4
votes
1
answer
133
views
Universal character ring for classical groups
The universal character ring for the general linear group is well understood but I want to ask about the universal character ring for the symplectic and orthogonal groups. For the general linear group,...
- 2,552
6
votes
0
answers
176
views
Functions of polynomial growth on linear algebraic groups
$\DeclareMathOperator\GL{GL}$Let $G$ be a complex linear algebraic group, i.e. a subgroup in $\GL_n({\mathbb C})$, defined by a system of polynomial equations
$$
p_i(x)=0
$$
(here $p_i$ are ...
- 7,426
4
votes
0
answers
147
views
Is the homogeneous coordinate ring of a flag variety a UFD?
I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
- 201
3
votes
1
answer
269
views
A more precise description of conjugation of semi-simple subgroups
Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
- 14.2k
4
votes
2
answers
499
views
density in SU(2,1)
Let $K=Q(\sqrt{-3})$ , is $SU(2,1)(K)$ dense in $SU(2,1)(C)$ for the complex topology?
- 14.2k
7
votes
1
answer
221
views
Is a quotient of real linear algebraic groups always a Cartesian product of compact and contractible factors?
Let $ G $ be the real points of a linear algebraic group and $ G' $ a Zariski closed subgroup. Then is $ G/G' $ a Cartesian product
$$
(K/K') \times F
$$
where $ F $ is contractible? Here $ K,K' $ ...
- 4,081
4
votes
1
answer
228
views
Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?
Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group.
I ...
- 12.9k
1
vote
1
answer
383
views
$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $
$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO.
Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
- 4,081
5
votes
2
answers
849
views
Stabilizers for nilpotent adjoint orbits of semisimple groups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (i.e. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...
- 63.9k
6
votes
1
answer
445
views
Is every finite subgroup the integer points of a linear algebraic group?
Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group?
Let $ ...
- 11.2k
4
votes
1
answer
257
views
Question regarding semistability of a point of GIT quotient
$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
- 166
4
votes
2
answers
332
views
Does the maximal compact subgroup always act transitively on a compact homogeneous space?
Let $ G $ be a Lie group, $ H $ a closed subgroup, and $ G/H $ compact. Under what conditions do we have that
$$
G/H \cong K/(K\cap H)
$$
where $ K $ is a maximal compact subgroup of $ G $? Obviously ...
- 4,081
2
votes
0
answers
69
views
Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 12.9k
3
votes
2
answers
263
views
Closed subgroups of $\operatorname{GL}_n(\mathbb C)$ with Lie algebra $\mathfrak{so}_n(\mathbb C)$
What is the classification of (Zariski) closed subgroups in $\operatorname{GL}_n(\mathbb C)$ (viewed as a linear algebraic group) with Lie algebra $\mathfrak{so}_n(\mathbb C)$?
Is it true that every ...
- 12.9k
1
vote
1
answer
345
views
Is the manifold of complex points of a quotient of compact groups just the tangent bundle?
In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
- 4,081
24
votes
2
answers
2k
views
Is it possible to realize the Moebius strip as a linear group orbit?
On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO:
Is the Moebius strip a linear group orbit? In other words:
Does there exists a Lie group $ ...
- 12.9k
2
votes
2
answers
213
views
Riemannian homogeneous equivalent to linear group orbit
Let $ M $ be a smooth manifold.
Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $.
Recall that a manifold $ M $ is Riemannian homogeneous if ...
- 4,081
2
votes
0
answers
127
views
Is the image of the exponential map of a complex semisimple group Zariski open?
Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map
$$\exp : \mathfrak{g} \to G$$
Zariski open in $G$?
- 443
23
votes
3
answers
2k
views
How bad can $\pi_1$ of a linear group orbit be?
Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
- 5,429
2
votes
1
answer
241
views
An extension of algebraic torus
Let $T_1$ and $T_2$ be algebraic tori over a field of characteristic 0. Let $T$ be an extension of $T_1$ by $T_2$, namely
$$
1\longrightarrow T_1\longrightarrow T\longrightarrow T_2\longrightarrow 1.
$...
- 12.9k
4
votes
1
answer
436
views
Universal covering groups of simple linear algebraic group schemes
Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
- 14.2k
6
votes
1
answer
1k
views
Centralizers of nilpotent elements in semisimple Lie algebras
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of $...
- 63.9k
4
votes
0
answers
93
views
Homomorphism from a product of spin groups to a bigger spin group
In the paper "Essential dimension of spinor and clifford groups" by Chernousov and Merkurjev, it says that there is a natural homomorphism
$\operatorname{Spin}(n)\times \operatorname{Spin}(m)...
- 321
6
votes
1
answer
679
views
Cartan decomposition of loop group
Let $G$ be a complex reductive group. Let $LG$ and $L^+ G$ denote the formal loop spaces given by maps from the punctured formal disk and the formal disk, respectively, to $G$. The quotient $LG/L^+ G$ ...
CommunityBot
- 1
4
votes
1
answer
701
views
Centralizers of semisimple subgroups
$\DeclareMathOperator\GL{GL}$If $G$ is a simple Lie group, and $\rho: G \to \GL(V)$ is a representation, then by Schur's lemma, the group of automorphisms of $\rho$ is a reductive subgroup of $\GL(V)$....
- 1,336
3
votes
0
answers
101
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
- 103
3
votes
0
answers
202
views
The group of fixed points of an involution of a Weyl group
Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...
- 14.2k
2
votes
1
answer
213
views
When is the symplectic group over a commutative ring generated by its root subgroups and a maximal torus?
This is related to Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups. There I was told that in general, the symplectic group $\text{Sp}_{2n}(R)$ is not generated by its ...
- 3,186
3
votes
1
answer
269
views
Symplectic group over $\mathbb{Z}/p\mathbb{Z}$ is generated by its root subgroups
This is a question about the answer in this other post: Symplectic group over integers and finite fields.
In general, for any ring $R$, the symplectic group $\text{Sp}(2n,R)$ is generated by its root ...
- 63.9k
4
votes
0
answers
180
views
Zariski density for certain subsemigroups
$\DeclareMathOperator\GL{GL}$Suppose that we have a Zariski-dense subgroup $\Gamma$ of $\GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series
$$
\sum_{x \in \Gamma} e^{-s \log\|...
- 89
4
votes
1
answer
1k
views
Complexification of compact Lie groups and complex algebraic linear reductive groups
I'm studying complexifications of compact Lie groups on "Representation of compact Lie groups- Dieck Brocker".
I found on internet that there is a bijection between complexifications of compact Lie ...
- 63.9k
5
votes
3
answers
2k
views
Centralizers of regular elements are abelian
Let $G$ be a complex semisimple Lie group with Lie algebra $\mathfrak{g}$. In a particular paper, the following statement is made:
If $X\in\mathfrak{g}$ is regular (i.e. has centralizer of minimal ...
- 4,279
1
vote
0
answers
91
views
Is this equivariant function constant?
Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
- 949
3
votes
0
answers
138
views
density of unipotent flows in algebraic groups
Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...
- 91
2
votes
1
answer
237
views
Properties of stabilizers of adjoint action general linear group
Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...
- 984
1
vote
0
answers
133
views
What is the analogue of Leibniz's rule for universal enveloping algebra?
Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra.
Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...
- 1,759
5
votes
0
answers
92
views
Canonical parabolics vs Levi subgroups
Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn ...
- 3,799
2
votes
0
answers
50
views
Centralizers of completely reducible subgroups
Let $k$ be a field of characteristic $p \geq 0$. Let $G$ be a connected reductive group defined over $k$ with $p$ good for $G$.
In what follows, I will cite results from the following two papers: ...
- 453
8
votes
2
answers
482
views
Parabolics and simple roots for a special unitary group: reference request
I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...
- 23k
27
votes
1
answer
3k
views
Definitions of real reductive groups
There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
The set ...
- 971
8
votes
2
answers
1k
views
Action of symmetric matrices under $\mathrm{O}(n)$
$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\...
- 21.5k
3
votes
1
answer
284
views
Schubert cells in G/P for reductive G
All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
- 44.7k
17
votes
4
answers
1k
views
Is $O_n({\bf Q})$ dense in $O_n({\bf R})$?
I am wondering if the orthogonal group $O_n({\bf Q})$ is dense in $O_n({\bf R})$?
It is easily checked for $n = 2$ but I think that there is a general principle concerning compact algebraic groups ...
- 108k
3
votes
0
answers
190
views
Harmonic analysis on reductive groups over $\mathbb{R}$
A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
- 131
12
votes
1
answer
392
views
Non-conjugate subgroups that are conjugate in complexification
In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H_1, H_2 \...
- 63.9k
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 : $ADB^{T}$, ...
- 35.5k
5
votes
1
answer
420
views
Analogue of the special orthogonal group for singular quadratic forms
The special orthogonal group $SO(n)$ is the subgroup of the special linear group $SL(n)$ of $n\times n$ matrices with determinant one that preserve a non-degenerate symmetric bilinear form. If such a ...
- 63.9k
2
votes
1
answer
137
views
Representation variety in $\mathrm{SU}(p,q)$
$\DeclareMathOperator\SU{SU}$Let $\Gamma$ be a cocompact oriented Fuchsian group, and consider the representation variety $\textrm{Hom}(\Gamma, \SU(p,q))$. Consider a point $\rho : \Gamma \to \SU(p,q)$...
- 63.9k
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 63.9k