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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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5
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1answer
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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0answers
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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1answer
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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0answers
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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1answer
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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2answers
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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0answers
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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1answer
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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0answers
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 ...
5
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1answer
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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0answers
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 ...
13
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1answer
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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0answers
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
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1answer
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 ...
2
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0answers
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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1answer
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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1answer
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 ...
6
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0answers
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 ...
3
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1answer
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 $\pi : G \to U(\mathscr{H})$ ...
6
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1answer
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 ...
2
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0answers
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 ...
5
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0answers
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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1answer
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
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1answer
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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1answer
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 $\Omega$ ...
10
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0answers
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 ...
2
votes
2answers
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
votes
1answer
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
votes
1answer
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 ...
2
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0answers
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 $\mathbf G$ ...
3
votes
1answer
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, ...
1
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1answer
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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0answers
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 ...
7
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1answer
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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0answers
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
2answers
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 $...
1
vote
1answer
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 ...
3
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1answer
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 $\{...
4
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1answer
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 ...
1
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1answer
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: $...
1
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0answers
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 ...
1
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1answer
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. ...
2
votes
1answer
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
1answer
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.

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