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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Haar Measures of $U(d)$ and $O(d)$

I'm trying to compute the Haar measures of finite unitary group $U(d)$ and finite orthogonal group $O(d)$. I've shown that both are compact groups. Most of computings I've seen base on Peter-Weyl ...
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15 views

What are some topics in representation theory of locally compact groups? [closed]

I am currently studying representations of locally compact groups and I find it a really interesting subject so I would like to know more. First I would like to ask if this is an active area of ...
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1answer
31 views

Reference for $p$-adic Haar integral

I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. ...
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1answer
38 views

Defining left Haar measure on locally compact group

Let $G$ is a locally compact group that is homeomorphic to an open subset (say $U$) of $\Bbb R^d$ ,and let $\varphi$ be a homeomorphism of $G$ onto $U$. Show that if for each $a$ in $G$ the function $...
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Isometrically embedding $L^1(H)$ into $L^1(G)$

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$. Denote $\mu_G$ as Haar measure on $G$ and $\mu_H$ as Haar measure on $H$. Is it possible to embed $L^1(H)$ into $L^1(G)$? By Radon-...
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1answer
68 views

Can we show $\mu(gX) = \mu(X)$ if integral is translation invariant?

Let $\mu$ be a Radon probability measure on the compact Hausdorff topological group $G$ such that $$\int_G f(g) \mu(dg) = \int_Gf(hg) \mu(dg)$$ for all $h \in G$ and for all $f \in C(G)$. Can I ...
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Explicit description of Barr's *-autonomous category of topological groups

In his paper On duality of topological groups, Micheal Barr describes a certain *-autonomous full subcategory $\mathcal{S}$ of the category of abelian topological groups. The groups that make up the ...
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1answer
47 views

The Dirac measure as a weak limit of $L^2$ functions on a LCA group.

Let $K$ be a locally compact abelian group. In the proof of Proposition 2 (the proposition does not matter for my question) of this blog-post, Tao writes: $K$ comes with an invariant probability ...
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$G$-invariant metrics on homogeneous spaces $G/H$

Let $G$ be a compact metrizable group and let $H$ be a closed subgroup. Let $d$ be a compatible metric on $G$ that is left-invariant and it is also right invariant with respect to the elements of $H$. ...
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26 views

Radon product of Haar measure is a Haar measure

Suppose we have a family of compact topological groups $\{G_\alpha\}_{\alpha \in A}$, define $G=\Pi G_\alpha$, it's a compact toplogical group equipped with coordinatewise multiplication and product ...
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1answer
52 views

Linear representations of compact groups

I was reading about linear representations of compact groups in the book of Jean-Pierre Serre: "Linear Representations of Finite Groups", and he states that most of the properties for representations ...
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1answer
25 views

How exactly is the Fourier transform the same as the Gelfand transform?

This answer states the Fourier transform is the Gelfand transform on the Banach algebra $L^1(G)$ with convolution. I've read the resource linked in the answer, but I still have some confusion. My ...
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1answer
26 views

are upper right triangular matrices a locally compact group?

Let $$G=\left\{\begin{pmatrix}a & b\\0& 1\end{pmatrix}:a>0,b\in \mathbb{R}\right\}$$ be endowed with matrix multiplication and the inherited topology as a subspace of $\mathbb{R}^4$. ...
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2answers
36 views

The Pontryagin dual of a compact abelian group is discrete

Suppose $G$ is a compact abelian group. Show that the Pontryagin dual $\hat{G}$ is discrete. I came up with this exercise when I read an introduction to Fourier analysis on locally compact abelian ...
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1answer
27 views

Discrete subgroup must be closed?

The question is : Let $G$ be a locally compact topological subgroup and $H$ is a discrete subgroup of $G$ (i.e. the hereditary topology of $H$ is discrete). Is $H$ always closed? I think the answer ...
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Free topological groups and quotients

I am learning about free topological groups, and I am trying to understand whether the analogue of "every group is a quotient of a free group" holds in the continuous setting too. I am particularly ...
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2answers
45 views

Is the topology on compact connected Lie groups metrizable?

The question is in the title, really. Suppose $G$ is a compact connected Lie group. Is there a metric on $G$ which induces the underlying topology? (so in particular $G$ is compact and connected wrt ...
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1answer
47 views

Group Convolution is Associative

Let $G$ be some locally compact group and $\mu$ its associated Haar measure. I am trying to adapt this proof that convolution on the locally compact group $(\Bbb{R},+)$ is associative. Here's what I ...
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1answer
45 views

Finite-dimensional compact groups

Let $G$ be a compact connected metrizable group of finite topological dimension. Is $G$ a Lie group? Or what should be equivalent in this case, does it have a faithful finite-dimensional ...
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1answer
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Statement is true for all continuous functions with compact support, then it must also be true in $L^{1}(G)$ by density?

I'm reading a proof about the existence of a left invariant mean on certain types of groups, and I came across this part that I don't quite understand: Let $G$ be a locally compact Hausdorff group. ...
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30 views

Can invariant metrics on quotients of groups $G/H$ always be obtained from invariant metrics on $G$?

Let $d$ be a (left-)invariant and compatible metric on a compact group $G$. Let $H$ be a closed subgroup of $G$. Then one can define on the homogeneous space $G/H$ an invariant and compatible metric $...
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22 views

Convergence on locally compact groups with an additional condition

This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto ...
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Does algebraic isomorphism of unitary irreps of a locally compact group imply unitary isomorphism?

Let $G$ be a locally compact group, $X_1,X_2$ unitary irreps of $G$. Assume that there is a $G$-module isomorphism $\phi:X_1\to X_2$ (not necessarily continuous). Does this imply that $X_1\cong X_2$ ...
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1answer
119 views

Understanding the roles of rapid decay & smoothness in the Fourier transform

1) If we define a rapidly decaying function in the usual way, it says nothing about derivatives; rather, just that its decay beats any polynomial growth. 2) A Schwartz class function is then simply a ...
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1answer
38 views

Defining the complexification of a representation

I am reading a paper and do not really understand the following. The setting is $G$ is a compact connected Lie group and $\mathbb{R}^n$ a representation of $G$. Later on in the paper the authors ...
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33 views

$\sigma$-Compactness--Is this Union Actually Countable?

Let $G$ be a locally compact group. The union of countably many open $\sigma$-compact subgroups of $G$ generates an open $\sigma$-compact subgroup. For each $n \in \Bbb{N}$, let $L_n = \bigcup_{j=1} ...
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17 views

Symmetric, Relatively Compact Open Neighborhoods of $1$

Let $G$ be a locally compact topological group. If $V$ is a symmetric, relatively compact open neighborhood of $1$, why is it true that $\overline{V}^n \subseteq V \cdot V^n$ for every $n \in \Bbb{N}$?...
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35 views

On the structure theory of the group of rational points of an algebraic group defined over a local non-Archimedean field

Let $k$ be a non-Archimedean local field, and let $G$ be an affine algebraic group over $k$. The literature suggests that the following are well established facts: The group $G(k)$ of $k$-rational ...
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20 views

Extension of Hilbert Inequality for locall compact groups

The classical Hilbert inequality states that if $f,g\in L^2(0,+\infty)$, then $$\int_0^{\infty}\int_0^{\infty}\frac{f(x)g(y)}{x+y}dxdy\leq\pi\|f\|_2\|g\|_2.$$ Are there analogs to the inequality for ...
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33 views

Orthogonality of $L^2(G)$ for a compact group

I am having trouble with the following: Let $G$ be a compact group with Haar measure $\mu$ and $(\rho,\mathbb{C}^n)$ be an irreducible representation of $G$. Naturally, we can suppose that $\rho$ is ...
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23 views

Proof for Tannaka–Krein duality?

I would like to find a proof for the Tannaka–Krein duality theorem as stated in the Wikipedia page. The usual reference for this is Hewitt & Ross, Abstract Harmonic Analysis II, but I find the ...
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1answer
15 views

An example of locally compact connected group which is not perfect

I would like to find an example of locally compact connected group $G$ which the commutator subgroup $G$ is not $G$ itself.
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1answer
138 views

Representation theory, orthogonality

I am trying to get my head around representation theory and was hoping for some help. I will write out some text and then ask questions and make comments after. Let $G$ be a compact connected Lie ...
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1answer
94 views

definition of unimodular group

Suppose $G$ is a locally compact group,we call $G$ an unimodular group if the modular function $\Delta =1$,that is to say,if the left Haar measure is also a right Haar measure. How to show that "$\...
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42 views

Question about regular representation of compact group.

I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
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1answer
31 views

Can the Fourier transform witness complete normality of the dual group?

Let $\Gamma$ be a countable Abelian group and let $G$ be its dual (compact) group. Consider the Fourier transform $F$ as a map from $\ell_1(\Gamma)$ into $C(G)$. Take a non-empty open subset $U$ of $...
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30 views

Naive questions on 1-cocycles of loally compact groups.

Let $\pi$ be an orthogonal representation of locally compact group $G$ on a real Hilbert space $H$. Recall that a continuous mapping $b \colon G \to H$ such that $b(gh) = b(g) + π(g)b(h)$, for all $g, ...
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49 views

The absolute convergence of (say) $\sum\limits_{n \in \mathbb Z} f(x+n)dx$ when $f$ is continuous and in $L^1(\mathbb R)$

Here is a basic theorem from locally compact topological groups. Suppose $G$ is a locally compact abelian topological group with closed subgroup $H$, and Haar measures $dg$ and $dh$. Suppose that $f:...
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61 views

Does $S^n$ admit expansive homeomorphism for some $n \ge 3$?

I know that $S^1$ and $S^2$ admit no expansive homeomorphism. But I don't know for higher dimension $S^n$ for some $n\ge3$. Definition: Let $X$ be locally compact metric space and $T$ is ...
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21 views

What are all invariant random subgroup on Sub ($R^n$) and Sub($G$)?

Let $G$ be a locally compact group. The space Sub($G$) be the closed subgroup of $G$ and $\mu$ be a borel probability measure of Sub($G$) which is invariant under all inner automorphism. What are ...
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2answers
53 views

For a connected semisimple Lie group $G$, $\mathrm{Inn}(G)$ is a subgroup of finite index in $\mathrm{Aut}(G)$.

For a connected semisimple Lie group $G$, $\mathrm{Inn}(G)$ is a subgroup of finite index in $\mathrm{Aut}(G).$ Can anyone give reference or proof for it. I know reference for compact but in general ...
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82 views

Prove that translation invariance implies inverse invariance without using the uniqueness of the measure

From the last paragraph of the proof for Claim V.11.8, [1], we know that Lemma. There is a unique regualr positive Borel measure $m$ on a compact group $G$ such that $$m(g\Delta)=m(\Delta)=m(\Delta ...
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1answer
77 views

Do every infinite locally compact Hausdorff group has infinitely many closed subgroup?

Do every infinite locally compact Hausdorff group has infinitely many closed subgroup? I know that every locally compact compactly generated abelian group is topologically isomorphic to $\Bbb R^n\...
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36 views

Modular function and left Haar measure

In what follows, $G$ is a locally compact group with left Haar measure $\lambda$ and modular function $\Delta$. I came across this statement : It follows from a careful application of Hölder's ...
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19 views

Is the left regular representation WOT closed?

Is the left regular representation of a locally compact group closed in the weak operator topology? If G is compact, by continuity, clearly the image is compact, therefore closed. However, in the ...
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1answer
60 views

Why is $\prod_{p\in\mathbb{P}}\mathbb{Z}/p\mathbb{Z}$ not an open subgroup of a compactly generated simple group

In my lecture we were given the following theorem and corollary. Theorem: Let $G$ be a compactly generated, topologically simple group. Then the $lpc(G)≠\emptyset$ and $lpc(G)$ is finite. (where $...
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2answers
69 views

Integration over a compact group with respect to Haar measure

Let $G$ be a compact abelian group and $\hat G={\rm hom_{\rm continuous}}(G,\mathbb{C}^*)$. Let $\mu$ be the Haar measure on $G$. For $\chi\in\hat G$, show that $$\int_G\chi\cdot\mu=1\quad\text{if}\...
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2answers
61 views

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
45 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
25 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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