All Questions
23 questions with no upvoted or accepted answers
6
votes
0
answers
201
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
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_{\...
- 2,071
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(...
- 4,600
5
votes
0
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140
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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 ...
- 474
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
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
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 $...
- 41
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 ...
- 359
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 ...
- 1,846
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 ...
- 185
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 ...
- 4,081
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,...
- 2,147
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)=
...
- 332
2
votes
0
answers
212
views
Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
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 ...
- 10.5k
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 $\...
- 14.2k
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
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(\...
- 332
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 \...
- 199
1
vote
0
answers
508
views
on the open bruhat cell
Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de Concini-...
- 3,472
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 $n = 5$...
- 376
0
votes
0
answers
168
views
semisimple conjugacy classes over general bases
Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if $\gamma,\gamma'\...
- 3,472
0
votes
0
answers
272
views
minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472