All Questions
Tagged with profinite-groups rt.representation-theory
20 questions
4
votes
0
answers
116
views
Are there Mackey decompositions for closed subgroups of profinite groups?
$\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}$
My question: what is known or expected to be true about the existence of a Mackey decomposition formula in the case of closed subgroups of ...
- 1,335
4
votes
0
answers
110
views
Hecke algebra $\mathcal{H}(K_1\backslash \mathrm{GL}_n(\mathbb{F})/K_1)$
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
- 21.8k
2
votes
1
answer
238
views
For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?
Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$...
- 769
4
votes
2
answers
367
views
An infinite profinite group such that any $p$-adic representation has finite image
Fix a prime $ p $. We call an infinite profinite group $ G $ a Fontaine-Mazur group (with respect to $ p $) if every continuous homomorphism $ G\to {\rm GL}_n(\overline{\mathbb{Q}}_p) $ has finite ...
- 863
6
votes
1
answer
558
views
Irreducible representations of product of profinite groups
It is a standard fact in the representation theory of finite groups that for $G,H$ finite groups, all of the irreducible representations of $G \times H$ are the external tensor product of irreps of $G$...
- 721
4
votes
1
answer
310
views
Confusion with self-dual representations of $\mathrm{GL}_n$ over a $p$-adic field
The following surely is kind of a trivial question, but it keeps me confused. It concerns a detail in Lust and Stevens' paper "On depth zero L-packets for classical groups" London Math. Soc. ...
- 769
2
votes
0
answers
82
views
On exactness of associating smooth representation-functor $(\,)^\infty$
Let $G$ be a locally profinite group, e.g. reductive group over $\mathbb{Q}_p$. For a (abstract) representation $(\pi,V)$ of $G$ and $K\subset G$ compact open subgroup denote by $V^K\subset V$ the $\...
- 473
4
votes
0
answers
102
views
What do the eigenvalues of a random element of $\mathbb Z_\ell[\Gamma]$ look like?
Let $\Gamma = \varprojlim \Gamma_n$ be a profinite group with $\Gamma_n$ finite quotients. For concreteness, let us fix $\Gamma_n = \operatorname{PGL}_2(\mathbb Z/\ell^n)$ so $\Gamma = \operatorname{...
- 7,736
6
votes
2
answers
570
views
Mackey theory in the setting of locally profinite groups
$\DeclareMathOperator\Hom{Hom}$Let $R$ be a commutative ring (not necessarily unital). Let $G$ be a finite group, and let $H_1, H_2$ be subgroups of $G$.
Recall the following standard result [1, Thm. ...
- 353
9
votes
0
answers
315
views
Colimit of continuous cohomology over subgroups
Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
- 2,197
9
votes
0
answers
371
views
Duality between coalgebras and (pseudocompact) algebras - uniqueness
The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below.
Let $k$ be a field. The ...
- 2,450
2
votes
0
answers
64
views
Representations theory of Groups with compact quotient
Let $G$ be a locally profinite group and $H$ a closed normal subgroup of $G$, with $G/H$ a profinite group.
If $\rho$ is an irreducible (smooth) representation of $G$ what can we say about the ...
- 335
12
votes
1
answer
427
views
Applications of Lubotzky's linearity theorem?
Lubotzky's theorem is a necessary and sufficient set of conditions for a finitely generated discrete group to be linear, i.e. isomorphic to a subgroup of $GL_n(K)$, where $K$ is a field of ...
- 26k
2
votes
1
answer
437
views
Finite quotients of an infinite product of finite groups
Let $G$ be a finite group.
Consider the direct product $\Gamma=\prod_{i=1}^{\infty}G$ of (countably) infinitely many copies of $G$. For every finite set of numbers $\{i_1,\ldots,i_n\}$ we have the ...
- 5,039
3
votes
1
answer
324
views
Faithful representations of free pro-p groups
Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \...
- 11.3k
3
votes
0
answers
816
views
Actions and representations of profinite groups
Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = \mathbb{Z}_p^{...
- 11.3k
5
votes
3
answers
1k
views
Action of a profinite group
Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
- 11.3k
6
votes
2
answers
984
views
Irreducible representations of compact groups
Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ${\...
- 11.3k
16
votes
3
answers
716
views
An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?
In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
15
votes
1
answer
656
views
Linear embeddings of nilpotent pro-$p$ groups
Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
- 657