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6 votes
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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
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
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
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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
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
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
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
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
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
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
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
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
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 ...
prochet's user avatar
  • 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 ...
naf's user avatar
  • 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 $\...
Mikhail Borovoi'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
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
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
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-...
prochet's user avatar
  • 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 $n = 5$...
A. Gupta's user avatar
  • 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'\...
prochet's user avatar
  • 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 ...
prochet's user avatar
  • 3,472