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Questions tagged [locally-compact-groups]

Locally compact groups are among the main objects of study of abstract harmonic analysis. They arise e.g. in number theory, abstract Fourier analysis, representation theory, Lie theory, operator theory, etc. Their main property is the existence and uniqueness of Haar measure. Please combine with relevant other tags whenever appropriate in order to reflect the main intentions of the question in the tags.

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Group action on convex cone

I was wondering if I could get some help understanding the following fact in an academic paper. The setup is as follows: An open subset of $\Omega \subset \mathbb R^k$ is an open convex cone if it ...
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1answer
37 views

Some notational question related to Haar measure (what do $d g^{-1}$ or $d (hg)$ mean?)

I am looking at some brief introduction to Haar measures and since I'm not understanding basic notion, I would greatly appreciate any clarification. Let $G$ be a locally compact group and say we ...
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1answer
16 views

Are conjugacy classes of compact groups Borel?

Let $G$ be a compact Hausdorff topological group. Is it necessarily the case that every conjugacy class of $G$ is Borel? It's certainly true if $G$ is countable (in which case $G$ is actually finite ...
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1answer
100 views

Prüfer Groups and Product Topologies

For each $p\in\mathbb{P}$, the Prüfer group $\mathbb{Z}(p^{\infty})$ is a divisible abelian group which can be given the discrete topology or the topology it inherits via its identification as a ...
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0answers
28 views

Non Compact Amenable group does not have Property (FH)

I'm learning from some notes about Analytic Group Theory and I'm stuck on an exercise there. Prove that a non compact Amenable group does not have Property (FH). At this point of the notes we ...
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Representation of locally compact groups

It is well-known that every compact Hausdorff group is the inverse limit of compact Lie groups (in fact, compact linear groups). Question: is there any analogue for locally compact (or at least lcsc) ...
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1answer
51 views

prove that any central function of $SU_{2}$ is uniquely determined by its restriction to the following subgroup.

The question is given below: But I do not know how to solve it, could anyone give me a hint please? EDIT:
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1answer
25 views

A linear topological space over real number field is locally compact ???

Let $V$ be a topological $\mathbb{R}$-vector space with ${\rm dim}(V) < \infty$ Then, $V$ is locally compact $??$
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1answer
71 views

Is it true that the Haar covering numbers $(f:\varphi)$ converges to $\int_Gf\operatorname{d}\mu$ for $\varphi\to\delta$?

Let $(G,\cdot,\tau)$ be a topological group whose topology is Hausdorff and locally compact and whose identity is $e.$ Denote by $\mathcal{B}_\tau$ the family of Borel subsets of $(G,\cdot,\tau)$, i....
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1answer
37 views

Is a locally compact subgroup of a locally compact group closed?

I encountered the following exercise in Terry's Tao book on Hilbert's fifth problem: Let $G$ be a locally compact group and let $H$ be a subgroup of $G$. Show that $H$ is closed if and only if $H$ ...
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2answers
55 views

Generated group by a compact set is finitely generated

Let $G$ be a locally compact group, and let $K\subseteq G$ be a compact set containing the identity. I want to make sure if the following statements are correct: (i) Is it true that $\langle K\...
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0answers
22 views

Understanding Gelfand–Raikov theorem

The context is unitary representations of locally compact topological groups. Theorem (Gelfand–Raikov). Let $G$ be a locally compact topological group. Then $G$ is separated by its irreducible ...
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41 views

Representations of $SO(3)$ in $\mathbb{C}[x,y,z]$

So my professor has given me the problem of examining the representations of $SO(3)$ in $V = \mathbb{C}[x,y,z]$. The thing is that he's now gone away for a conference trip, and I have absolutely no ...
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0answers
37 views

About Locally Compact Groups and Analysis

This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed. Let $G$ be a locally compact group and let $f \in C_c(G)$ where $C_c(G)...
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0answers
51 views

When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?

Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
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0answers
54 views

Can 3 dimensional Heisenberg group be represented irreducibly on L^2(S)?

It is well-known that unitary dual of the 3 dimensional Heisenberg group H represented on $L^2(\mathbb{R})$ is given by a nonzero real number $\lambda\in R^*$(can be interpreted as $1/\hbar$). When $\...
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0answers
52 views

Let $\text{T} = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$

I have a question about algebraic tori, let $T = \mathbb{Q}(i)^\times/\mathbb{Q}^\times$ and let's try to compute $\text{T}(\mathbb{A})/\text{T}(\mathbb{Q})$. One has a product decomposition for ...
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2answers
58 views

Is a finitely generated metrizable group discrete?

The question is in the title. A countable locally compact Hausdorff group is discrete, so saying that a finitely generated metrizable group is locally compact would be enough. What if the group is a ...
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2answers
182 views

Is the following set a compact set?

Let $A$ be defined as $$A:=\{f\in C^1([0,1],\mathbb{R}) : \|f\|_{C^1} \leq 1\}.$$ I have shown that the set is precompact. But is $A$ a complete set? Or an other question: Is $A$ a closed set?
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0answers
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How does the locally compact group play a role in many branches of math? [closed]

It seems that Fourier analysis/harmonic analysis plays an important role in math, at least in number theory and statistics. It also seems to me that talking about it is talking about locally compact ...
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1answer
38 views

Definition of Haar integral in Bushnell and Henniart

In Bushnell and Henniart's The Local Langland's Conjecture for GL(2) they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional $$ I: C^{\infty}_{c}(G) \...
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0answers
35 views

Dimension of vector space for irreducible representations

Let $G$ be a locally compact group, $\rho$ an irreducible unitary representation on some inner product space $V$. Is there a bound on the dimension of $V$ with respect to the cardinality of the group? ...
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1answer
72 views

The collection of (unitary representations on) Hilbert spaces is a set

Let $G$ be a locally compact group. I know that the collection of all unitary representations of $G$ is not a set, since there are unitary representations on inner product spaces with bases of any ...
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1answer
25 views

Reference request for locally compact groups proof

Where can I find a proof of the following: "Every locally compact group is a directed union of $\sigma$-compact open subgroups" This is claimed in Greenleaf's book but I haven't been able to find a ...
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1answer
30 views

Locally compact nilpotent group has an open subgroup isomorphic to $\mathbb{R}^n\times K$

My question is about a possible generalization of the following structure theorem of locally compact abelian groups. Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact ...
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0answers
18 views

Correspondence of Rep($G$) and Rep($\mathfrak{g_\mathbb{C}}$) for a compact Lie group $G$.

Let $G$ be a real compact Lie group, $G_\mathbb{C}$ its complexification, $\tilde{G}$ its universal cover, and $\mathfrak{g}$ be its Lie algebra. I have seen in lots of occasions that the following ...
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1answer
111 views

Is the family of equivalent norms a locally compact space?

Consider an infinite dimensional Banach space $(X,||\cdot||)$. Let $\mathcal{P}$ be the family of all equivalent norms in $X$. That is $p\in \mathcal{P}$ iff $p$ is equivalent to $||\cdot||$, i.e. ...
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0answers
18 views

A question about a mapping between spaces of continuous functions in LCGs

In Raghunathan's book Discrete subgroups of Lie groups we have in the chapter Generalities on Lattices the following setting: $G$ is a Locally compact group, $H$ a closed subgroup of $G$, and $G,H$ ...
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0answers
37 views

Laplace operator on certain groups

Q1. What are some canonical ways to define Laplace operator on a discrete group? in particular if $f\in \mathcal L^2(\mathbb Z)$ how is $\Delta f$ defined? Aside considering Laplacian on graphs, are ...
3
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1answer
44 views

How to prove that if $f*\chi_A=0$ a.e. for all $A$ of finite measure then $f=0$ a.e.

Let $G$ a locally compact abelian group and let $\mu_G$ an Haar measure of $G$. Suppose that $f\in L^1(\mu_G)$ is such that for all measurable $A$ of finite $\mu_G$-measure it happens that $f*\chi_A=0$...
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0answers
26 views

Local compactness and actions

Let $X$ be a Hausdorff topological space and $G$ a group acting continuously on $X$. We denote the group action as $(g,x) \rightarrow gx$. Let $x\in X$ and $U$ a neighborhood of $x$ such that the set $...
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0answers
36 views

Integration of representations of a locally compact group

I am studying Alain Robert's Introduction to the Representation Theory of Compact and Locally Compact Groups, where I found a notion that might be generalized. Given a locally compact group $G$ and a ...
2
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1answer
93 views

If $\prod X_{\alpha}$ is locally compact $\Rightarrow$ each…

If $X:= \prod X_\alpha$ is locally compact $\Rightarrow$ each $X_\alpha$ is locally compact and $X_\alpha$ is compact for all but finitely many values of $\alpha$ My solution So I will use that if $f:...
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0answers
31 views

List of locally compact abelian groups

I wish to have a lot of examples of locally compact abelian groups. I cannot find a big list, so perhaps we can make on here. Of course, the Fourier transform yields a duality. Hence given a locally ...
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0answers
24 views

Does Pontryagin duality extend to class two nilpotent or maybe even metabelian locally compact groups?

A metabelian group $G$ is determined up to isomorphism by "abelian data" $(A,M,\alpha)$ -- an abelian group $A:=G/[G,G]$, an $A$-module $M:=[G,G]$, and a cocycle $\alpha:A\times A\to M$ giving the ...
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1answer
42 views

Subset sums in an LCA group

Let $G$ be LCA, compact, $F\subseteq G$, $F+F=G$, $F$ open. Is it true that for each $g\in G$, $g+F\cap F\neq \emptyset$? In particular, this is equivalent to $F-F=G$. I've tried a bunch of examples ...
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1answer
43 views

Dual group of $\mathbb{R}\setminus\lbrace 0 \rbrace$

I want to calculate the dual group of $(\mathbb{R}\setminus\lbrace 0 \rbrace, \cdot)$. By assuming $\chi(x)=\exp(2\pi i f(x))$, I concluded from the character property $\chi(xy)=\chi(x)\chi(y)$, that ...
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2answers
79 views

Connected locally compact abelian groups

Is the category $\text{LCA}_c$ of connected locally compact Hausdorff abelian groups an abelian category? My feeling says no, however I can't immediately find a counterexample. Alternatively, I'd ...
3
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1answer
39 views

$\Big(f\in A(G)$ & $H(x) = f(x^2) \Big){\Longrightarrow}^{?} H\in A(G)$

Let $G$ be a locally compact topological group and $A(G)$ be the Fourier algebra. For $f\in A(G)$ we define $H(x) = f(x^2)$. Could we say that $H\in A(G)$? What about discrete group $G$? For ...
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0answers
30 views

Conjugation in Hausdorff topology group

Let $G$ be a first countable Hausdorff topological group and $g\in G\setminus \{e\}$. Denote $[g]$ be the conjugacy class of $g$. My question is that "Can $e\in \overline{[g]}$?", where $\overline{[g]...
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0answers
38 views

topological vector space over locally compact field

Let $K$ be a nondiscrete locally compact topological field, and $V$ be a finite dimensional topological vector space over $K$ and $\{ v_1, \cdots ,v_n\}$ be a base of $V$. $\varphi : K^n \rightarrow ...
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174 views

Finite measure fundamental domain for a discrete group implies it's a lattice

Here $G$ is a locally compact second countable topological group with left haar measure $\mu$, and $\varGamma$ is a discrete subgroup with a borel subset $\varOmega \subseteq G$ s.t. $G=\biguplus_{\...
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1answer
107 views

Inverse map preserving positivity with Haar measure?

It is known that homeomorphisms in general do not preserve null sets. However, I am wondering about the case where $G$ is a locally compact group with haar measure $\mu$ and the homeomorphism in ...
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0answers
36 views

When does $0=\int f(x)\psi(x)\,dx$ for all $f\in L^1(G)$ imply that $\psi=0$

Let $G$ be a locally compact group with Haar measure $dx$. I am wondering what conditions a function $\psi$ must satisfy so that the following holds. $\int f(x)\psi(x)\,dx=0$ for all $f\in L^1(G)\ \...
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1answer
80 views

Relationship between the induced measure on an orbit and Hausdorff measure on the orbit

Assumptions Let $G$ be a locally compact group that acts continuously on $\mathbb{R}^n$. Let $\mu$ be a right Haar measure on $G$. Let $F:\mathbb{R}^n \to \mathbb{R}^m$ ($m < n$) be a Lipchitz ...
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1answer
66 views

Identities related to Schur's orthogonality relations.

In Jacques Faurat's book, Analysis on Lie Groups. An introduction, on page 105, he states the following theorem. Theorem 6.3.3. Let $\pi$ be an irreducible unitary $\mathbb{C}$-linear representation ...
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0answers
85 views

Contragredient of a cuspidal representation

Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal. A ...
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1answer
230 views

Convolution of a function and a measure.

Consider a locally compact group $\mathrm{G}$ and a left-invariant Haar measure $\lambda$ on it. Let $\mu$ be a probability measure. Suppose $f$ is a function continuous and bounded. Denote by $\Delta$...
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111 views

The measurability of convolution in locally compact group

Prove or disprove: Let $G$ be a locally compact Hausdorff topological group and $\mu$ be a left haar measure on $G$. $f\in L^1(G,\mu)$, $g\in L^{\infty}(G,\mu)$. Then $f*g$ is measurable with respect ...
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0answers
36 views

Every quasicharacter of an open subgroup extends to a quasicharacter of the group

Let $H$ be an open subgroup of a locally compact Hausdorff abelian group $G$. Assume that $G/H$ is a finitely generated abelian group. Let $\chi: H \rightarrow \mathbb{C}^{\ast}$ be a continuous ...