# Questions tagged [unitary-representations]

The unitary-representations tag has no usage guidance.

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### Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ ...

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201 views

### tensor product of massless Poincare representations

Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless ...

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### Restriction that contains a trivial representation

Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$ such that $(G,G')$ is a symmetric pair. If $\pi$ is an infinitely dimensional unitary representation of $G$, ...

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307 views

### Finite-dimensional faithful unitary representations of SL(2,Z)

Does $SL(2,\mathbb{Z})$ have a finite-dimensional faithful unitary representation? No such representation exists for $SL(2,\mathbb{R})$, but I don't see a reason why one shouldn't exist for $SL(2,\...

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### Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices

Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I ...

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### Some basic question on the parabolic induction

I would like to ask some basic question about parabolic induction.
Let $F$ be a local field and $G=GL_n(F)$ and $P=MN$ its parabolic subgroup whose Levis subgroup $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...

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138 views

### Is the linear span of irrep matrices a complete matrix basis?

Let $G$ be a finite or compact group and $\rho: G \to \mathrm{U}(d)$ a $d$-dimensional unitary representation of $G$. If $\rho$ is irreducible then the following seems to be true:
$$
\mathrm{span}_\...

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54 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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151 views

### Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?

The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of $G$ ...

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31 views

### Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications? [duplicate]

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is ...

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51 views

### Number of orthogonal operators in representations of the Unitary Group

Let $G={\rm SU}(d)$ be the unitary group and $\rho(g)$ an irreducible representation of $g\in G$ in a $D$ dimensional Hilbert space $V$. Let $e_i\in V$ be the diagonal matrix whose only non-zero ...

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50 views

### Fourier transform in the complex motion group

I am looking for a reference that deals with the dual of the complex motion group i.e., the semi-direct product of $\mathbb C^n$ with the special unitary group $K=SU(n)$. In particular, I am looking ...

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182 views

### Discrete decomposability of unitary representation

[INTRODUCTION]
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...

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190 views

### Systems of imprimitivity for unitary representations - reference request

Let $G$ be a finite subgroup of the group $U_d(\mathbb{C})$ of unitary transformations of $\mathbb{C}^d$. Suppose that $G$ acts irreducibly but is imprimitive, meaning that there is a nontrivial ...

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168 views

### Unitary dual of the motion group $M(n)$, for $n> 2$

The motion group of $\mathbb R^2$, noted by $G=M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well
known fact is that the unitary dual $\hat{G}$, of ...

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99 views

### When is the unitary dual of a lscs group uniformizable?

Let $G$ be a locally compact, second countable group. We equip the unitary dual $\widehat{G}$ with the Fell topology. I am looking for conditions which guarantee that the topological space $\widehat{G}...

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### Tensor products of unitary irreducible representations of $SU(2,2)$

What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...

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### sum of singular values of diagonal transformations of a unitary matrix

Let $\mathbf{D}_1=\text{diag}(d_1,...,d_n)$ and $\mathbf{D}_2=\text{diag}(d_1',...,d_n')$ be positive diagonal matrices. Let $\mathbf{U}$ be a unitary matrix ($\mathbf{U}\mathbf{U}^t=\mathbf{U}^t\...

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86 views

### Collection of matrices in $SU_{\mathbb{C}}(n)$ with given family of eigenvectors

For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?
If we consider a fixed set of $n$ complex vectors $\Gamma:=\{...

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108 views

### Existence of a unique cyclic and separating vector in a *-representation

I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...

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64 views

### Existence of a unique state-representation relation in a C*-algebra

My question is whether there exists a unique cyclic and separating state on a C$\ast$-algebra giving rise to a specific representation (in other words is it possible to reverse the GNS construction)? ...

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141 views

### Does the supercuspidal representation becomes tempered?

I am really wondering whether supercuspidal representation may become tempered representation.
If it is not true for all classical group, is it especially true for unitary group?
If it is not true ...

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### relationships between $AA^T$ and $[(I-A)(I-A)^T]^{-1}$ with $A$ being strictly lower triangular

I have a matrix $A$ which is strictly lower triangular. Now, I am trying to find some general statements/relationships of following matrices $U,D,V,K$ defined as:
$AA^T=VKV^H$,
$[(I-A)(I-A)^T]^{-1}=...

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### Good source for representation of GL(n) over finite fields?

I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated.
======== edit =========
My original question was ambiguous. ...

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149 views

### Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces

I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...

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### the moment set of unitary representation of lie groups, analogue in the p-adic case

Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$ and $\pi$ a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_{\pi}$. Note $\mathcal{H}_{\pi}^{\infty}$ the space of $...

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145 views

### Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...

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### The relation of the local principal representations of $U(2)$ and $GL(2)$

Let $E/F$ be a quadratic extension of number fields and $v$ is a non-archimedean place of $F$.
Let $G=U(2)(F_v)$ be the $F_v$-points of the 2-dimension unitary group associated to $E_v/F_v$ and $B$, $...

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### Operators associated with unitary representations of nilpotent Lie group

Let $G$ be a nilpotent Lie Group, and $\pi:G\to B(\mathcal H)$ be an irreducible unitary representation on the Hilbert space $\mathcal H$. One can use the Bochner integral to define a linear map as ...

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### Principal series representations of $SL(2,\mathbb{R})$: introductory textbooks [duplicate]

I am interested in introductory books/papers/reports about the (unitary) representation theory of $SL(2,\mathbb{R})$, with particular emphasis on the principal series representations. My background: I ...

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### Two Hilbert $G$-bundles are isomorphic iff the representations of the little group are equivalent

While reading Mackey's "Unitary Group Representations in Physics, Probability and Number Theory", at page 66 I encountered the following statement (my reformulation):
if $\mathcal K$ and $\mathcal ...

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### Plancherel formula for $L^2(G/N)$

Let $G$ be a connected real semisimple or reductive Lie group. Let $TA$ be a Cartan subgroup, where $T$ is compact and $A$ is split. Let $MA$ be the centralizer of $A$ in $G$, and let $N$ be the ...

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### When can a finite subgroup of $GL(2n,\mathbb{R})$ be viewed as a subgroup of $GL(n,\mathbb{C})$?

A finite group acting on a complex vector space of dimension $n$ can be seen as acting on a real vector space of dimension $2n$ just by forgetting the complex structure of the space. My question is, ...

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### Restriction of irreducible unitary representation to normal subgroup of finite index

Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...

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### Unitary representations of SO(1,4) and SO(2,3)

Where can I find details about the irreducible unitary representations of SO(1,4) and SO(2,3)?

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### non unitary representations

A paper by M.L. Whippman in Rep. Math. Phys. 5 (1974), 81, mentions at the bottom of the second page ''the possible occurrence of non-unitary representations that arise when reducing a direct product ...

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### Orthogonality relations for unitary representations of infinite (finitely generated) groups

Let $G$ be a group, and consider the matrix elements of finite dimensional irreducible unitary representations of $G$ over $\mathbb{C}$ as functions $f:G\to \mathbb{C}$. If $G$ is finite, any two ...

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487 views

### A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite.
I do not think the ...

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### Are square-integrable representations of a general locally compact group tempered

Let $G$ be a locally compact unimodular group with center $Z$ and let $\omega$ be a unitary character of $Z$. To fix the discussion, an irreducible unitary representation $\pi$ with central character $...

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### Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...

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### Characters of permutation groups

Let $N$ be a fixed positive integer, and denote by $C(m)$ the number of
permutations on an $N$-element set that have exactly $m$ cycles (counting
$1$-cycles). Then it is in the literature that the ...

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### Orbits of some action of SL2 on Pontryagin dual of the field of formal Laurent series

Let $K=\mathbb{F}_2((t))$ be the field of formal Laurent series over the finite field $\mathbb{F}_2$. Now consider $K^3$ as an additive group and its dual group $\hat{K^3}$, which consists of all ...

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### references for faithful orthogonal (or unitary) representation of symmetric groups

Let $S_n$ be the symmetric group of $n$ points. I want to find references (or proofs) for the following statement (1).
(1). There does not exist any faithful orthogonal representation
$$
S_n\...

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390 views

### Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which
1- is finitely generated by $S$,
2- does not have property (T),
3- admits infinitely many finite quotients which do not factor through an homomorphism $G \...

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426 views

### On the definition of matrix coefficient

As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:
For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient ...

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### Are there any dominant pivotal functors such that the regular representation is not mapped onto a multiple of the regular representation?

This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.
A tensor functor $F: \mathcal{C} \to \mathcal{D}$ is called dominant (sometimes ...

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### Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations?

I'm asking a question about Lie group representation.
Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...

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### Unitarizability of group representations

Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...

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### The representation theory for the fake Heisenberg groups over non-perfect local field

Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
G=\{\begin{...

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100 views

### Number of unitary representations of a Kazhdan group

It was proved by de la Harpe, Robertson, and Valette that for a discrete group $\Gamma$ with Kazhdan's property (T), there is a constant $c$ so that the number of irreducible unitary representations ...