1
vote
2answers
106 views

When is the induced representation factored through the initial one?

Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ ...
2
votes
0answers
134 views

A decomposition of a representation via characters of a normal compact subgroup

This is connected to my question here. Let $K$ be a normal compact subgroup in a locally compact group $G$, $\widehat{K}$ the dual object for $K$, and $\mu_K$ the normed Haar measure on $K$ ...
5
votes
3answers
306 views

Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...
3
votes
1answer
91 views

Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$. Do we have a analog of Siegel subset for the quotient ...
1
vote
1answer
256 views

When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
0
votes
1answer
120 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
2
votes
2answers
229 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...
2
votes
1answer
117 views

The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations, Theorem 6.1 (Selberg Trace formula) on page 26 of these notes. Equation 3.19 and 3.20 on page 11 of this paper. I vaguely feel that these two are the ...
8
votes
0answers
179 views

when do norm-continuous unitary representations separate points of a group?

Recently I found in the web a discussion on the following question: ...
3
votes
4answers
337 views

How does one show the existence of discrete and complementary series for SL(2,R)?

In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation ...
10
votes
3answers
359 views

Topology on the Unitary Dual

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...
3
votes
1answer
153 views

Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...
1
vote
0answers
163 views

Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
8
votes
2answers
770 views

Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e. $$ \theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ . $$ I'm ...
5
votes
0answers
148 views

variant of Haar measure

I found a certain trig identity in a discussion on Lie groups \[ \frac{\prod_{i < j} 2 \sin (\mu_i - \mu_j) \prod_{i< j} 2 \sin (\nu_i - \nu_j) }{\prod_{i, j} 2 \cos (\mu_i - \nu_j) } = ...
4
votes
2answers
251 views

“geometric” description of the algebra of central functions on a Lie group

I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
2
votes
1answer
251 views

Characters separating points on Maximal Torus modulo Weyl group?

Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group. Every finite-dimensional representation of G has a character, which is a function on G, T and ...
5
votes
1answer
219 views

What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$?

I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex ...
5
votes
2answers
404 views

Is there any way to generalize the Laplacian to finite groups?

The group theoretic interpretation of harmonic analysis was born out of the observation that the discrete Fourier transform on a signal of length $n$ was precisely the Fourier transform of the finite ...
7
votes
2answers
784 views

Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality). The basic algorithm is ...
2
votes
3answers
863 views

Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups. Let $G$, $H$ be two reductive groups defined over a number field $F$. Let $\mathcal{A}(G)$ be ...
10
votes
3answers
1k views

Is there a Plancherel Theorem for Gowers norms?

In the process of counting arithmetic sequences in sets, the Gowers norms $$ ||f||\_{U^s[N]}^{2^s} = \frac{1}{N^s} \sum_{\vec{h} ,\\, n } \Delta_{h_1}\dots\Delta_{h_s}f(n) $$ where the sum is $ ...
11
votes
4answers
1k views

Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...
1
vote
1answer
127 views

Dominant weights appear in Discrete Series

If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...
2
votes
1answer
444 views

Measures and structure on conjugacy classes

Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$ $$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} ...
4
votes
1answer
536 views

Multiplicity of eigenvalues of the Laplacian on quaternionic projective space

Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-1}{k-1}$, ...
3
votes
1answer
230 views

Inducing from cocompact subgroups

Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of ...
7
votes
1answer
794 views

First eigenvalue of the Laplacian on Berger spheres

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the ...
9
votes
3answers
929 views

Why is the dual of a torus the same as its fundamental group?

The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...
7
votes
2answers
569 views

Decomposing an arbitrary unitary representation of a connected nilpotent Lie group in terms of its irreps

For a locally compact (Hausdorff) abelian group $G$ we have following theorem (see e.g. Folland): "For every (strongly continuous) unitary representation $(\pi,\mathcal{H_{\pi}})$ of $G$, there ...
20
votes
3answers
1k views

When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?

I was writing up some notes on harmonic analysis and I thought of a question that I felt I should know the answer to but didn't, and I hope someone here can help me. Suppose I have a compact ...
2
votes
3answers
601 views

Plancherel formula for special linear group

I am looking for a comprehensible material covering Plancherel formula for $SL(n,\mathbb{R})$ and $SL(n,\mathbb{C})$. Of course, I wouldn't mind reading an explanation for general semisimple Lie ...
11
votes
3answers
2k views

Harmonic analysis on semisimple groups - modern treatment

For my finals, I am digging through the book by Varadarajan An introduction to harmonic analysis on semisimple Lie groups. I find it a rather hard read and I feel it's a bit outdated now. Any ...