# 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.

124
questions

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130 views

### When does Haar measure decompose into products of such?

We get a "nice" Haar measure on $G=SL(2,R)$ in Iwasawa coordinates $G=NAK$ as follows: $dg=dx {dy\over y^2} dk$. Here $N=\{ n_x\}$, $A=\{a_y\}$ and $K=SO(2)$. Note that $dg=dn\, da\, dk$ is ...

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vote

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245 views

### Haar measure coming from Pontryagin duality v/s Fourier inversion

Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...

**5**

votes

**0**answers

158 views

### Haar mesure on $\mathrm{GL}_{d}(F)$

$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...

**0**

votes

**1**answer

146 views

### 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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**2**answers

219 views

### “Haar-like” measure on $S_\omega$

Let $S_\omega$ be the collection of bijections $f:\omega\to \omega$. Endow $\omega$ with the discrete topology and let $S_\omega$ be endowed with the subspace topology of $\omega^\omega$, where $\...

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67 views

### Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary.
To my surprise, it was never answered. So I am elevating it to MathOverflow.
I ...

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24 views

### Distribution on the inner product of random projections, concentration of measure

For given $d$, $I_d$ denotes the identity matrix.
Let $2d\times 2d$ matrix $P=\begin{pmatrix}
I_d & 0\\
0 & 0
\end{pmatrix}$.
Let us consider
$UPU^{+}=\begin{pmatrix}
A & B\\
C &...

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54 views

### Haar measure decomposition using orbital integrals

Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...

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76 views

### Haar measure and Integral

I am wondering whether the following integral over Haar measure has explicit form（edit: say $U$ is $d\times d$ unitary, orthogonal or symplectic matrix)
$$
\int dU [(U\otimes U^*)X(U^{+}\otimes U^T)]^{...

**7**

votes

**1**answer

185 views

### A group where the Weil topology induced by the Haar measure does not coincide with the original topology

Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-...

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vote

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105 views

### Pushforward of measure under surjective map

Let $X, Y, Z$ be measurable spaces with measures $\mu_X, \mu_Y, \mu_Z$ respectively. Let $\pi_Y : Y \times Z \rightarrow Y$ be the projection on $Y$ and $\pi_Z : Y \times Z \rightarrow Z$ the ...

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97 views

### Is this a lattice?

Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...

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37 views

### How to prove invariance implies quadrature about Haar measure?

I'm reading paper, which contains a lemma named as 'Invariance Implies Quadrature'. The lemma is stated as follows.
Lemma Let $f$ be a function from $O(d)$ to $\mathbb{R}$ and let $H$ be a finite ...

**5**

votes

**1**answer

139 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 ...

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vote

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97 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**

votes

**1**answer

151 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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91 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**

votes

**1**answer

239 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 $\...

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votes

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408 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**

votes

**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 $\alpha$ ...

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votes

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201 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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92 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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56 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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96 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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votes

**1**answer

114 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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107 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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113 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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169 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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votes

**1**answer

458 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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134 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 $\...

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votes

**1**answer

186 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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118 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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87 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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196 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) \...

**0**

votes

**1**answer

115 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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70 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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votes

**1**answer

67 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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votes

**1**answer

376 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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97 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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186 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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115 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}{...

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**1**answer

444 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

363 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 $\...

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239 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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286 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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votes

**2**answers

787 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 ...

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votes

**1**answer

270 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 ...

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**1**answer

173 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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230 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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99 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, ...