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Questions tagged [unitary-representations]

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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 $\...
Adrián González Pérez's user avatar
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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 ...
benkj's user avatar
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Fourier transform in the complex motion group

I am looking for a reference that deals with the unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$ i.e., the semi-direct product of $\mathbb C^2$ with the special unitary group $K=...
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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 ...
Monty's user avatar
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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 ...
Changguang's user avatar
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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 ...
M.B's user avatar
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Isometric representation semisimple?

The first lemma on p.35 of these notes states that unitary representations are semisimple. Could the same be said of isometries if the space doesn't have an inner product? This topic notes that the ...
Adam Hughes's user avatar
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Unitary representations of the symmetric group over finite fields

I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant Unitary representations of finite groups over ...
Jackson Walters's user avatar
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Linear algebraic group, absolute root system, computing roots

Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
user536406's user avatar
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Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...
Andrew's user avatar
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Density of irreducible matrix coefficients of a locally compact group

Let $G$ be a locally compact group and $I$ the set of matrix coefficient of irreducible unitary matrix coefficients of $G$. By Gelfand-Raikov's theorem and Stone-Weirestrass's theorem, for a compact $...
Pople's user avatar
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Classifying endomorphisms of a direct sum Hilberts pace

Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$...
Sam Makhoul's user avatar
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Are generalized symmetric groups maximal finite groups (in a certain sense)? - Part II, Loose Ends

Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
Jonas Anderson's user avatar
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Bounding the dimensions of faithful representations of a quotient group

For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...
rick's user avatar
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A p-adic analogue of a result due to Kirillov

Let $k$ be a non-Archimedean local field with char$(k)=0$. Let $N$ be the group of $k-$rational points of a unipotent algebraic group defined over $k$. It is known that $N$ is a locally compact and ...
Pedro A. Matos's user avatar
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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 $$...
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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 $U(H)$ ...
Hugo Chapdelaine's user avatar
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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 ...
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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}=...
user3093264's user avatar
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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 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{...
m07kl's user avatar
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Representation of finite group

Let $\Gamma$ be a finite index subgroup of $SL_2(\mathbb{Z})$. Let $\pi$ be a natural covering from $\Gamma\backslash\mathbb{H}$ to $SL_2(\mathbb{Z})\backslash\mathbb{H}$. Denote by $\text{Deck}(\pi)$ ...
TNT's user avatar
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Do the irreducible unitary representations of a locally compact group form a separating set for the Radon measures on the group?

Let $\mu$ and $\nu$ be two Radon measures on a locally compact group $G$. For every irreducible unitary representation $\pi$ of $G$ and vectors $u$ and $v$ from the corresponding Hilbert space $H_\pi$ ...
bla's user avatar
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About the group generated by one diagonal unitary

Suppose $D=diag\{\alpha_1,\alpha_2,...\alpha_n\}$ is a diagonal unitary, which means that |\alpha_i|=1 for all $i$. We know that $\alpha_i$ is not unit root and so is $\alpha_i/\alpha_j$ for $i\neq j$....
gondolf's user avatar
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Sampling orthogonal matrices from Haar-random unitary group

I would like to know the probability of sampling orthogonal matrices $O \in O(d)$ from Haar-random unitary group $U(d)$. The probability may be close to zero since orthogonal matrices are "sparse&...
Chris H's user avatar
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A reference for this statement (representations of universal central extensions)

Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact: "Every projective unitary ...
Mahtab's user avatar
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A question on projective unitary representation of a Lie group

$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
Mahtab's user avatar
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How to build a representation of the diffeomorphism group of $U(n)$?

Given that $U(n)$ is a smooth manifold I would like to know if there is a way of building a representation of $\text{Diff}(U(n))$ once you pick a particular (finite dimensional) representation of $U(n)...
Nicolas Medina Sanchez's user avatar
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Tempered representations and unramified principal series

For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...
InteresetingStuff's user avatar
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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 ...
m07kl's user avatar
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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 ...
asv's user avatar
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faithful representation of locally compact group

I have been thinking about existence of faithful representation of locally compact groups. This representation exists for example for compact lie groups. But I am curious to know if one can say some ...
Mohammad's user avatar

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