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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questions with no upvoted or accepted answers
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Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
20
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301
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
12
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200
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Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?
So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
11
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258
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Status of the analog of the Haar measure on quantum groups
In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...
10
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427
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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 ...
7
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621
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Concentration of measure for uniform distribution on Stiefel manifolds
This is my first post on MO, so I hope the question is suitable. I am looking at the uniform distribution on the Stiefel manifold, but more specifically, at the uniform distribution on the ...
6
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87
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The uniform odd and even subgraph of $\mathbb{Z}^2$
Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
6
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239
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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,...
6
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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 ...
5
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187
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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
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
5
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183
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Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
5
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212
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On generically Haar-null sets in the real line
First some definitions.
For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
5
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125
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Volumes of Hecke operators
Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets
$$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...
5
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315
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The Haar integral on uniform spaces
Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...
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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 ...
4
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Haar measure on Galois groups
Galois groups are nice compact Hausdorff groups, and therefore possess a bounded Haar measure, unique if we insist that the total volume be $1$. What is the Haar measure on the absolute Galois group ...
4
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712
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Haar measure on strictly upper triangular matrices
Let F be a function field, and A its adele ring. I want to consider U(A)/U(F), where U(A) is the space of strictly upper triangular matrices with entries from A, and U(F) is the same with entries ...
3
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112
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Integrating over a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action and the choice of the fundamental domain
Let $\mathcal{F}$ be a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action. It is well known that there exists a unique $\text{SL}(d,\mathbb R)$-invariant probability ...
3
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1
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Maximum norm within a random subspace intersected with an ellipsoid
Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$.
Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define
$U(a) = \{u \in \mathbb{R}^n: \...
3
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0
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140
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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)
$$
\int dU [(U\otimes U^*)X(U^{+}\otimes U^T)]^{...
3
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0
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Is this a lattice?
Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...
3
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156
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Question about regular representation of compact group
I first define the setting for my question. Let $G$ be a compact group with probability Haar measure $\mu_G$. Denote by $\lambda$ the left regular representation on $L^2(G)$ defined for $f \in L^2(G)$ ...
3
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Conditional distributions of uniformly distributed random orthonormal matrices
Let $U, U'\in R^{d\times k} (d>k)$ be two independent uniformly distributed random orthonormal matrices. In specific, let $S$ be the set of all $d\times k$ orthonormal matrices. Here 'uniform' is ...
3
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710
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Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
3
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0
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270
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Chen's iterated integrals and free loop space
I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one ...
3
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0
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271
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Almost conjugation-invariant neighborhoods of units in locally compact groups
Let $G$ be a locally compact topological group with unity $e$ and left Haar measure $m$. Let also $g\in G$ be a given element and $U$ a neighborhood (of compact closure) of $e$.
I am interested to ...
3
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0
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104
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Haar Functionals and Coquasi-triangular Structures
In this question it is mentioned that the coordinate algebra $C_q[G]$ Drinfeld--Jimbo algebras, for $G$ a compact semi-simple Lie group, admit a unique positive definite Haar functional. I was ...
2
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108
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Double quotient integral formula on $\Gamma \backslash G /K$
Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...
2
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Formulation of $p$-adic Haar measure decomposition
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\vol{vol}\DeclareMathOperator\diag{diag}$Suppose:
$F$ is a non-archimedean local field,
$\mathcal{O} \subset F$ its ring of integers,
$\pi \in \mathcal{...
2
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0
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80
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Biased ensemble in the unitary group
I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows
$$
\mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2}...
2
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148
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About the probability of satisfying in a commutator type equation
Let $N$ be a closed normal subgroup of a compact group $G$. We denote the unique Haar measure of a compact group $A$ by $\mathbf m_A$, and drop $A$ if there exists no ambiguity. Fix $y\in G$. For $g\...
2
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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 $\alpha$ ...
2
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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 $\...
2
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0
answers
136
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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 $T$ ...
2
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102
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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 ...
2
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0
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398
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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 $\mathbf G$ ...
2
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0
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69
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Differential of Haar integral function
Let $C^k$ action of a compact Lie group $G$ on $R^m$, $D(g)$ denote the differential of the map $x\in R^m \mapsto g(x) \in R^m$ at the origin and $\mu$ is the normalized Haar mesure on $G$, consider a ...
1
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0
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265
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Using the von Neumann crossed product to introduce a measure on the orbit space?
Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question: is there a natural way of using the ...
1
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0
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167
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Integration with respect to Haar measures normalised over a subspace
Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...
1
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0
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97
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Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure
In this question, the following fact was used by the respondent
A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar
measure contains a coset of $G^0$, the connected component of
$G$ ...
1
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0
answers
110
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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 \...
1
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0
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206
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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 $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
1
vote
0
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93
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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 ...
1
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0
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188
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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 ...
1
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0
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265
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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 ...
1
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0
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139
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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 ...
1
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0
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127
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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 $...
1
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0
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85
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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 ...
1
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0
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418
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Question on the Quotient Integral Formula
I have a very concrete question on the proof of the following (see below): Given a 'nice' top. space $X$ and a 'nice' group operation of $G$, say, from the right, on $X$ and a certain measure on $X$, ...
1
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0
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219
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exotic compact group
Let $G$ be compact (and Hausdorff) group, $\mu$ be Haar measure on $G$. Is it always true that $(G,\mu)$ is a standard probability space (Lebesgue-Rokhlin space)? It is so if (a priori not iff) the ...