# Questions tagged [haar-measure]

Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.

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### Asymptotics of Haar moments on general Lie groups

I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
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### Problem concerning Haar measures on locally compact Hausdorff groups [migrated]

Before I state what my problem is I first want to give some context. A Haar measure is a measure on the Borel subsets of a locally compact Hausdorff group $X$. The Haar measure is inner regular on ...
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### Haar measures of compact subgroups

Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$: $$\mu_K(K)=1.$$ Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
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### Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case. The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
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### Calculation of first correction to Selberg type integral

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix. $\Tr U$ will denote the character ...
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### Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold?

I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such ...
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### Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic

I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
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### Haar measure on ${\cal P}(\omega)$

First, we note that there is a natural bijection ${\cal P}(\omega) \to \{0,1\}^\omega$ and endow the latter with the product topology (where $\{0,1\}$ carries the discrete topology). So we get a ...
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Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set $$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^... 4 votes 1 answer 305 views ### Measure of subsets of profinite groups Let G be an infinite profinite group, so$$G=\lim_{\longleftarrow}G/N$$where N runs through the open normal subgroups. I have two questions: Is G of Haar measure zero in the compact group \... 8 votes 2 answers 588 views ### Average of the maximum matrix element over the Haar measure Let U be a d\times d unitary matrix, and U_{i,j} be its matrix elements. I am interested in the following quantity$$\int dU \max_j |U_{1,j}|^2 \ ,  where $dU$ is the uniform Haar measure over ...
Here is the setup. I have a submersion $f:X \to T$, where $X$ is manifold and $T$ is a torus (you can chose a circle for the beginning if it is simpler). The manifold $X$ has a volume form $\alpha$ ...