Skip to main content

All Questions

Filter by
Sorted by
Tagged with
9 votes
2 answers
865 views

Multiplication in Peter-Weyl theorem

$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
Yellow Pig's user avatar
  • 2,964
4 votes
1 answer
171 views

Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?

Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
Another User's user avatar
6 votes
1 answer
352 views

All surjections onto trivial irrep split equivalent to being reductive

$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations $$ 0 \to W \to V \to k \to 0 $$ ...
Ian Gershon Teixeira's user avatar
8 votes
1 answer
534 views

Representation theory of $\mathrm{GL}_n(\mathbb{Z})$

I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
Kenji's user avatar
  • 81
1 vote
0 answers
151 views

On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$

$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...
naf's user avatar
  • 10.5k
2 votes
0 answers
65 views

Are the integer points of a simple linear algebraic group 2-generated?

Set Up: Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
Ian Gershon Teixeira's user avatar
15 votes
2 answers
613 views

Existence of a regular semisimple element over $\mathbb{F}_{q}$

This is probably old, a Chevalley level of old, but I'm not at all an expert in this field so I need help. Let $G$ be a simply connected (almost) simple linear algebraic group defined over $K=\mathbb{...
D. Dona's user avatar
  • 455
4 votes
2 answers
181 views

The orbits of an algebraic action of a semidirect product of a unipotent group and a compact group are closed?

We consider real algebraic groups and real algebraic varieties. It is known that the orbits of an algebraic action of a unipotent algebraic group $U$ on an affine variety are closed. The orbits of an ...
Vladimir47 's user avatar
3 votes
0 answers
237 views

Centralizers and algebraic groups

Suppose $G$ is a linear algebraic group - I am also happy to assume $G$ is a simple algebraic group over an algebraically closed field of characteristic zero, but the question won't require this. The ...
James Freitag's user avatar
1 vote
0 answers
97 views

A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori

$\newcommand{\Hom}{{\rm Hom}} \newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}} \newcommand{\X}{{\sf X}} $ I am looking for a reference for the following lemma (for which I know a proof): Lemma. Let $\...
Mikhail Borovoi's user avatar
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, ...
GTA's user avatar
  • 1,024
6 votes
1 answer
255 views

Which Lie groups are a central extension of an algebraic group?

Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
Luis's user avatar
  • 161
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 ...
Ian Gershon Teixeira's user avatar
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.,...
stupid_question_bot's user avatar
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)...
YJ Kim's user avatar
  • 321
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\|...
Zestylemonzi's user avatar
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 ...
BharatRam's user avatar
  • 949
2 votes
1 answer
238 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 ...
Tommaso Scognamiglio's user avatar
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. ...
Mikhail Borovoi's user avatar
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 $\GL(V)$....
unknownymous's user avatar
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 ...
user avatar
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_{\...
Andrew NC's user avatar
  • 2,071
0 votes
0 answers
99 views

Unimodular matrices fixing $(1, 1, \cdots, 1)$

What is known about the subgroup of $GL(n, \mathbb Z)$ fixing (under left multiplication) the vector ${(1, 1, \cdots, 1)}^T$ ('T' denotes transposition). I'm particularly interested in the case $n = 5$...
A. Gupta's user avatar
  • 366
8 votes
1 answer
337 views

What is the subgroup of $\mathrm{SL}(n,\mathbb{C})$ which preserves the discriminant?

$\DeclareMathOperator{\SL}{\operatorname{SL}}$Let $\mathcal{P}_{n-1}$ be the space of complex polynomials in one variable, say $z$, of degree at most $n-1$. As a complex vector space, it is clearly $n$...
Malkoun's user avatar
  • 5,215
5 votes
0 answers
123 views

Conjugacy classes of plane k-jet group

Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
Denis T's user avatar
  • 4,600
2 votes
0 answers
290 views

Automorphisms group of complex and real simple Lie algebras

$\DeclareMathOperator{\Inn}{\operatorname{Inn}}\DeclareMathOperator{\Aut}{\operatorname{Aut}}\DeclareMathOperator{\Out}{\operatorname{Out}}\DeclareMathOperator{\g}{\mathfrak{g}}$According to Wikipedia,...
annie marie cœur's user avatar
11 votes
2 answers
935 views

Non-isomorphic complex Lie groups with the same exceptional Lie algebra for $\mathfrak{g_2,f_4,e_6,e_7,e_8}$?

An exceptional complex Lie algebra is a simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly five such Lie algebras: $\mathfrak{g}_{2}$, ${\mathfrak {f}}_{4}...
annie marie cœur's user avatar
5 votes
1 answer
729 views

The normalizer of block diagonal matrices

Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the ...
Francois Ziegler's user avatar
3 votes
0 answers
71 views

Conjugacy classes in reductive group under adjoint action of parabolic subgroup

Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
Simon Lentner's user avatar
4 votes
0 answers
68 views

The weak restriction of the Jacquet module

Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
IMED's user avatar
  • 41
3 votes
0 answers
160 views

Orbit representatives for the action of the maximal compact subgroup

Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...
user8974's user avatar
  • 185
6 votes
1 answer
658 views

Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$. Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$. Let $\sigma$ be an anti-...
Mikhail Borovoi's user avatar
2 votes
0 answers
173 views

Characterize an element of $\operatorname{SL}_n(\mathbb Z)$

I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$. In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where $$e_{1,n}(m)= ...
Ami's user avatar
  • 332
1 vote
0 answers
370 views

Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$

Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$? For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
Ami's user avatar
  • 332
3 votes
1 answer
276 views

For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?

Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
Ami's user avatar
  • 332
2 votes
1 answer
216 views

How to prove that Chevalley groups over $\mathbb R$ have no compact factors

I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem. I've been told in another thread ...
Ami's user avatar
  • 332
2 votes
1 answer
259 views

Character of a semisimple connected Lie groups [closed]

I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character? I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
Ami's user avatar
  • 332
7 votes
1 answer
467 views

Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups

I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
Ami's user avatar
  • 332
4 votes
1 answer
119 views

Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields

The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis. In ...
Arkandias's user avatar
  • 991
5 votes
0 answers
140 views

Intermediate subgroups between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, for anisotropic form of $SL_2$

Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$. Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups ...
JadeSnail's user avatar
  • 474
3 votes
2 answers
1k views

Examples of groups for which Margulis superrigidity theorem applies

I am not an expert at all in the subject of Lie groups, lattices, arithmetic groups and rigidity. But, lately I am interested in Margulis superrigidity theorem, which in most versions can be stated as ...
Luis Jorge's user avatar
4 votes
1 answer
350 views

A generalization of Siegel property

In reduction theory of arithmetic groups, one has the following finiteness property. Proposition 1 (Siegel property). Let $G$ be a reductive group over $\mathbb{Q}$ and let $\Gamma\subset G(\mathbb{...
Golden Wave 's user avatar
2 votes
1 answer
156 views

About the conjugation of reductive subgroups

I am considering the following question. Question 1. Let $G$ be a reductive algebraic group over $\mathbb{R}$, can we find finitely many reductive $\mathbb{R}$-subgroups $H_1,...,H_m$ such that for ...
Golden Wave 's user avatar
1 vote
0 answers
289 views

A question about Mostow's theorem for self-adjoint groups

In Mostow's paper Self-adjoint groups, one can find the following property. Theorem. Let $G\subset \mathrm{GL}_{n,\mathbb{R}}$ be a reductive real algebraic subgroup. Then there exists $a\in \...
Golden Wave 's user avatar
7 votes
2 answers
231 views

Non-semisimple symmetric subgroups of simply connected simple algebraic groups

Let $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\...
Mikhail Borovoi's user avatar
3 votes
2 answers
555 views

Counterexamples to Margulis Normal subgroup theorem in rank 1

Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index. What are the known counter-examples in rank $1$ ? I am ...
Xavier Roulleau's user avatar
4 votes
1 answer
633 views

Homomorphisms from binary polyhedral group to compact Lie groups

Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified? For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie ...
Yuji Tachikawa's user avatar
6 votes
1 answer
390 views

Can these two irreducible $GL_n \mathbb Z$-representations be isomorphic?

Fix $n\in \mathbb N$ and a partition $\lambda$ with at most $n-1$ parts (of length at most $n-1$). Let $V$ be the irreducible $GL_n \mathbb R$-representation with highest weight $\lambda$ and $D$ the ...
Peter Patzt's user avatar
7 votes
1 answer
442 views

Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
Jarek Kędra's user avatar
  • 1,782
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 $...
Golden Wave 's user avatar