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4
votes
1answer
126 views

Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. A function $\lambda\colon G\to \mathbb C$ is an ...
2
votes
0answers
32 views

Realization of limit of discrete series using Dirac operators

I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ...
3
votes
0answers
160 views

Kazhdan Property T of semisimple Lie groups

I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN). I want to ...
2
votes
0answers
58 views

Relation between topological and differentiable Lie group cohomology for unitary modules

It is well known, that if $V$ is a $C^\infty$ $G$-module, then the differentiable cohomology groups $H^*_{d}(G, V)$ are isomorphic to $H^*(\mathfrak{g}, K, V)$. I am interested if the result can be ...
1
vote
0answers
36 views

Examples of groups admitting a proper $1$-cocyle for a bounded representation

A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...
4
votes
0answers
92 views

Systems of imprimitivity for irreducible subgroup of GU(n,q)

My question is similar to this one but about finite field case. So, the set up is the following: Let $G$ be $GU_n(q)$ acting on unitary space $(V, {\bf f})$, where $V=\mathbb{F}_{q^2}^n$ and ${\bf f}...
4
votes
3answers
273 views

Parametrization of real-valued SU(N)

I want to construct a $SU(N)$ matrix $V$, with the following property: All the elements of the first row are given, i.e. $V_{1,j}=a_i$ (with $\sum_i a_i^2=1$) All matrix elements are real, i.e. $V_{i,...
3
votes
0answers
79 views

Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
1
vote
0answers
55 views

Commutation fo a self-adjoint operator with a unitary operator

Let $A$ be a selfadjoint bounded operator on a Hilbert space. Let $M$ be another bounded selfadjoint operator. Let me assume the commutation property $$ [A, e^{iM}]=0. \tag 1$$ Does (1) imply that $$...
8
votes
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 ...
1
vote
1answer
192 views

Unitary condition

I came across the following while doing some related proof; It seems easy to prove. $\quad$ We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$: $1$) Given a unitary $n\times n$ matrix $U$, there is ...
4
votes
1answer
172 views

Uniform Roe algebra of virtually abelian group is type I C*-algebra?

Let $G$ be an arbitrary (discrete) group. It acts by left translation on $\ell^\infty(G)$. The uniform Roe algebra of $G$ is defined as the crossed product $\ell^\infty (G) \rtimes_{\mathrm{red}}G$. ...
7
votes
1answer
173 views

K-type in discrete series representation

The following result seems well known. Let $G$ be a reductive Lie group with a maximal compact subgroup $K$. If $\mu$ is an irreducible unitary representation of $K$, then there exist only finitely ...
2
votes
1answer
99 views

Finding all unitary representations of the connected Poincaré group

I am studying representation theory of Lie groups and its combination to theoretical physics, and I am concerned about the following. Is there an exhaustive way to find all unitary representations of ...
1
vote
0answers
67 views

Irreducible unitary representations of discrete abelian groups

It seems to me that the statement below should be true but I would like to double-check. Statement: Let $H$ be a (separable) complex Hilbert space and consider its associated unitary group $U(H)$ ...
3
votes
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 $f \in L^2(G)$ ...
3
votes
0answers
89 views

Is the character of the adjoint representation of $\operatorname{SU}(n)$ non-vanishing on regular points of a maximal torus?

Are the maximal and minimal values of the character of the adjoint representation of $\operatorname{SU}(n)$ restricted to a maximal torus known? Can such a character vanish at some regular point of a ...
6
votes
2answers
272 views

Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$

I would like to know what all the possible finite-dimensional representations of the following relations are. $$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$ $$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\...
1
vote
0answers
52 views

Weyl theorem for non specified primitive root of unity

Let $\omega=e^{2i \pi/p}$. Weyl theorems give all representations of matrix algebra span by $A,B$ such that either $AB=\omega BA, A^p=B^p=I$, or $(k,l)\mapsto A^kB^l$ is a irreducible ...
2
votes
0answers
77 views

Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity

Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$. Define the $p$ fourrier transform ...
5
votes
2answers
252 views

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 $G$ ...
3
votes
1answer
248 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 ...
3
votes
0answers
66 views

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 $G$, ...
10
votes
1answer
436 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,\...
4
votes
1answer
606 views

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 ...
3
votes
0answers
125 views

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 $M=GL_{n_1}(F) \times GL_{n_2}(F)$ ...
2
votes
1answer
411 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}_\...
3
votes
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 $\pi : G \to U(\mathscr{H})$ ...
3
votes
0answers
166 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 $G$ ...
1
vote
0answers
36 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 ...
2
votes
0answers
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 ...
1
vote
0answers
52 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 ...
2
votes
1answer
196 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 ...
4
votes
1answer
277 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 ...
2
votes
1answer
181 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 ...
4
votes
1answer
120 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}...
7
votes
1answer
300 views

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 ...
1
vote
1answer
87 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:=\{...
3
votes
0answers
121 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_{\...
1
vote
0answers
162 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 ...
1
vote
0answers
40 views

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}=...
14
votes
5answers
1k views

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. ...
6
votes
0answers
172 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-...
3
votes
0answers
78 views

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 $...
1
vote
0answers
165 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 ...
3
votes
0answers
69 views

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 $B$, $...
2
votes
0answers
67 views

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 ...
3
votes
1answer
478 views

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 ...
3
votes
0answers
73 views

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 ...
5
votes
0answers
158 views

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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