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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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 ...
Michael_1812's user avatar
1 vote
0 answers
110 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 \...
darkl's user avatar
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3 votes
0 answers
140 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)]^{...
gondolf's user avatar
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8 votes
1 answer
291 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 $\sigma$-...
Saúl Pilatowsky-Cameo's user avatar
1 vote
1 answer
391 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 ...
Andrew Musso's user avatar
3 votes
0 answers
109 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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46 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 ...
Nate's user avatar
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6 votes
1 answer
273 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 ...
Dieter Kadelka's user avatar
1 vote
0 answers
206 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 $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
Sitan Chen's user avatar
3 votes
1 answer
315 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^...
MCS's user avatar
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0 answers
97 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}^...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
330 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 $\...
Meisam Soleimani Malekan's user avatar
8 votes
2 answers
619 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 ...
user149918's user avatar
2 votes
0 answers
59 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 $\alpha$ ...
Sovy's user avatar
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4 votes
1 answer
225 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 ...
MrsHaar's user avatar
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3 votes
0 answers
156 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 $f \in L^2(G)$ ...
Constantin K's user avatar
1 vote
0 answers
93 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 ...
Rick Sternbach's user avatar
5 votes
0 answers
183 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 ...
D_S's user avatar
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1 answer
234 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....
Gro-Tsen's user avatar
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0 answers
237 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 ...
B.Gillan's user avatar
1 vote
0 answers
188 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 ...
J.F's user avatar
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1 vote
0 answers
265 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 ...
Daniel Tausk's user avatar
13 votes
1 answer
556 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 $\...
Gro-Tsen's user avatar
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2 votes
0 answers
220 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 $\...
jmc's user avatar
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4 votes
1 answer
342 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 \...
John von N.'s user avatar
1 vote
0 answers
139 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 ...
D_S's user avatar
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2 votes
0 answers
136 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 $T$ ...
Paul Broussous's user avatar
1 vote
1 answer
225 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) \...
RobR's user avatar
  • 83
0 votes
1 answer
240 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 ...
Sellerie's user avatar
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6 votes
0 answers
81 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 ...
Jan_Ch.'s user avatar
  • 113
3 votes
1 answer
77 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 $\pi : G \to U(\mathscr{H})$ ...
Constantin K's user avatar
7 votes
1 answer
560 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 ...
user120980's user avatar
2 votes
0 answers
102 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 ...
Taras Banakh's user avatar
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5 votes
0 answers
212 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 ...
Taras Banakh's user avatar
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5 votes
0 answers
125 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}{...
Wolker's user avatar
  • 541
7 votes
1 answer
898 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) + \...
Jing-Yuan Chen's user avatar
7 votes
1 answer
655 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 $\...
gradstudent's user avatar
  • 2,136
8 votes
2 answers
357 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 $\Omega$ ...
Alexander Pruss's user avatar
10 votes
0 answers
427 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 ...
L.C. Ruth's user avatar
  • 229
5 votes
2 answers
1k 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 ...
D_S's user avatar
  • 6,100
3 votes
1 answer
410 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 ...
user's user avatar
  • 33
2 votes
1 answer
271 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 ...
D_S's user avatar
  • 6,100
2 votes
0 answers
398 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 $\mathbf G$ ...
D_S's user avatar
  • 6,100
3 votes
1 answer
134 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, ...
D_S's user avatar
  • 6,100
1 vote
1 answer
569 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 $m,k$ ...
gondolf's user avatar
  • 1,487
3 votes
0 answers
266 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 ...
Minkov's user avatar
  • 1,117
8 votes
1 answer
516 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 ...
Minkov's user avatar
  • 1,117
1 vote
0 answers
127 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 $...
Lior Bary-Soroker's user avatar
7 votes
2 answers
2k 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 $...
timofei's user avatar
  • 297
1 vote
1 answer
282 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 ...
nullUser's user avatar
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