# 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.

111
questions

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106 views

### Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group

Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables :
Let $X$ and $Y$ be $G$-valued ...

**1**

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**0**answers

75 views

### L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...

**3**

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**1**answer

117 views

### Asymptotic analysis using the p-adic Mellin Transform?

In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...

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88 views

### How large this subset is to say that it should equal the group?

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**

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**1**answer

208 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**

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**2**answers

338 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 ...

**2**

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**0**answers

52 views

### Approximating a volume along a submersion

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 $\alpha$ ...

**5**

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**1**answer

188 views

### Haar-null union of dense subsets

Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...

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67 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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55 views

### Haar measure of the zero set of a nonconstant analytic function on a connected Lie group

Let $G$ be a connected Lie group equipped with its unique real analytic structure, $f : G \to \mathbb{R}$ a nonconstant real analytic function on $G$. Is the closed set $Z_f = f^{-1}(0)$ always of ...

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74 views

### Invariant measure on coset space and integrable functions

Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...

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102 views

### The square modulus of coordinates of a uniformly chosen point in complex projective space is uniform in the simplex

I can't recall where I learned this (beautiful) fact, and I would like a reference (if possible, in a textbook):
Let $(z_0:\cdots:z_n) \in \mathbb{P}^n(\mathbb{C})$ be chosen uniformly at random w....

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76 views

### Restriction of a Haar measure

Let G be a locally compact topological group with Haar measure $d_G$, H be a compact subgroup of G with normalized Haar measure $d_H$ and N be the smallest normal subgroup of G containing of H with ...

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92 views

### Extrinsic applications of Haar measure

I am looking for examples of (not necessarily deep) results whose proofs rely on the Haar measure (or rather such that there exists an "elegant" proof involving it). Further, the formulation of such ...

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117 views

### Invariant measures on locally compact homogeneous spaces

Edit: answered in the first comment. This turned out to be really easy, as the orbits of an open $\sigma$-compact subgroup yield a partition of $X$ into open $\sigma$-compact subsets.
Let $G$ be a ...

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**1**answer

435 views

### Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...

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101 views

### Does the pushforward of the Haar measure of a semisimple compact Lie group along a character determine the character?

Let $G$ be a connected compact semisimple Lie group.
Let $V$ be a faithful representation of $G$,
with character $\chi \colon G \to \mathbb{C}$.
Let $\mu_G$ be the normalized left Haar measure. (So $\...

**4**

votes

**1**answer

162 views

### Invariant integration on principal bundles

Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \...

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115 views

### Do we have $K \cap P = (K \cap M)(K \cap N)$?

Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...

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78 views

### Volume of a double class of a parahoric subgroup

Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...

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vote

**1**answer

190 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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**1**answer

95 views

### Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...

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69 views

### Are invertible measures strictly dense?

Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...

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65 views

### Convergence of some object depending on functions with compact support

Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...

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335 views

### Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo ...

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92 views

### Is this concrete set generically Haar-null?

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete.
First we recall the definition of a generically Haar-null set in ...

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178 views

### On generically Haar-null sets in the real line

First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...

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111 views

### Volumes of Hecke operators

Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets
$$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...

**6**

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**1**answer

377 views

### Haar Measure Integral

I am a physicist and I am wondering whether the following integral over Haar measure （edit: say $U$ is unitary, orthogonal or symplectic matrix)
\begin{align}
\int dU \: \exp\left( \mathrm{tr}(UX) + \...

**6**

votes

**1**answer

305 views

### About taking an expectation over orthogonal matrices

Say $Q$ is a random variable which is sampling orthogonal matrices in $m$ dimensions using the Haar measure on $O(m)$. Let $A$ and $B$ be some (fixed) subset of rows and columns of $Q$ such that $\...

**7**

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**1**answer

218 views

### Extending the product measure on $2^\omega$

Consider the standard completed product measure $P$ on $\Omega=\{0,1\}^\omega$ corresponding to an i.i.d. sequence of fair coin-flips.
Given $n\in\omega$, let $\rho_n$ be the bijection of $\Omega$ ...

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269 views

### Haar measure on $PGL(2,\mathbb{Q}_p)$, the local Jacquet-Langlands correspondence, and Ihara's theorem

Goal. I would like to calculate the product of the formal dimension of a discrete series representation of $GL(2,\mathbb{Q}_p)$ with trivial central character (so, an irreducible unitary ...

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**2**answers

623 views

### Simple Proof that a Reductive Group is Unimodular?

Let $G$ be a connected, reductive group over a local field $k$ of characteristic zero. I thought of a simple proof that $G(k)$ is unimodular, but I realize it is almost certainly wrong: $G(k)$ is ...

**3**

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**1**answer

232 views

### Invariant measureable function is constant

I'm wondering whether the action of $\mathbb{Q}/\mathbb{Z}$ on $S^1$ by multiplication is ergodic.
Note that this question is equivalent to each one of the following two statements
Is every ...

**2**

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**1**answer

146 views

### How does a Haar measure on $N$ arise from root subgroups?

Let $G$ be a connected, reductive group, split over a local field $F$. Let $B = TU$ be a Borel subgroup defined over $F$ with maximal torus $T$ and unipotent radical $U$. Let $P$ be a parabolic ...

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180 views

### How can this argument calculating the Haar measure on a parabolic subgroup be generalized to the non-split case?

Let $\mathbf G$ be a connected, reductive group over a local field $F$. Assume there is a maximal torus $\mathbf T$ which is split over $F$. Let $\mathbf P$ be a parabolic subgroup of $\mathbf G$ ...

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**1**answer

96 views

### Measure on orbits of $N$ under conjugation by $H$

Let $G$ be a locally compact topological group with closed subgroups $H, N$ and $H$ normalizing $N$. Then $H$ acts continuously on $N$ by conjugation. If it will help, assume that $N$ is nilpotent, ...

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vote

**1**answer

376 views

### Integral over Haar measure

$\mathcal{H}$ is a $d$-dimensional complex vector space. $\mathcal{E}$ maps matrix on $\mathcal{H}^{\otimes m+k}$ to matrix on $\mathcal{H}^{\otimes m}$ through
$$\mathcal{E}(X)=EXE^{+},$$
where $m,k$ ...

**3**

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**0**answers

253 views

### Conditional distributions of uniformly distributed random orthonormal matrices

Let $U, U'\in R^{d\times k} (d>k)$ be two independent uniformly distributed random orthonormal matrices. In specific, let $S$ be the set of all $d\times k$ orthonormal matrices. Here 'uniform' is ...

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**1**answer

335 views

### Frobenius norm of the principal submatrix of a uniformly distributed random orthonormal matrix

Suppose that we have a uniformly distributed $d\times d$ random orthonormal matrix $\mathbf{X}$. Here "uniform" is defined in the sense of Haar measure, i.e., the distribution does not change up to ...

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113 views

### Proof/reference for a variant of Pontryagin duality

Let $X,X'$ be locally compact abelian groups with a non-degenerate quadratic form
$\left<\bullet ,\bullet \right>\colon X\times X' \to \mu_{l}$,
where $l$ is a prime, and $\mu_l$ the group of $...

**7**

votes

**2**answers

801 views

### Haar measure on the Grassmannian space

The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $...

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vote

**1**answer

217 views

### Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...

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**1**answer

64 views

### Criterion for convergence of sums for non-continuous functions

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{...

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**1**answer

236 views

### On the eigenvalues' distribution of random unitary

Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group.
For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...

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**1**answer

205 views

### $p$-adic Dirac measure as a weak limit

The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta_n(x)$ spiked at the the origin, in the sense that:
$...

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**0**answers

80 views

### Majorizing inequality on spectral norm of product of a random and a deterministic low-rank projection

Let $P$ be a rank $k$ uniformly randomly oriented projection matrix in ${\mathbb R}^d$ -- this is constructed as $R^T(RR^T)^{-1}R$ where $R$ is a $k\times d, k<d$ random matrix with i.i.d. 0-mean ...

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173 views

### Some question on haar measure for sumsets of closed subsets of profinite groups

Let $H$ be a profinite group with the haar measure $\mu_H$. Let $H_1$ and $H_2$ be closed subgroups of $H$. $H_1$ and $H_2$ have their own haar measures $\mu_{H_1}$ and $\mu_{H_2}$ respectively.
...

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votes

**1**answer

317 views

### Averages of vector inner products over the Haar measure

Consider arbitrary unit vectors $w,x,y,z \in \mathbb{C}^d$. Is there an explicit formula for what this average is?
$$
\int \mathrm{Tr}( \psi \psi^* \, \, w x^* \,\, \psi \psi^* \,\, y z^*) d\psi
$$
...

**2**

votes

**1**answer

213 views

### Subgroups of finite non-zero Haar measure of abelian locally compact groups

Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.