The unitary-representations tag has no wiki summary.
2
votes
1answer
205 views
Decomposing a reducible representation of the unitary group
Consider the representation $L_U$ of the unitary group $U(n)$ on $L(\mathbb{C}^n)$ where $L_U$: $L(\mathbb{C}^n) \rightarrow L(\mathbb{C}^n)$ is a linear operator that $L_U M=U M U^{\dagger} $, ...
2
votes
1answer
289 views
Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$
As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...
5
votes
0answers
124 views
Proof that the second Borel cohomology group of $(\mathbb R, +)$ is trivial
Does anyone have a reference for a fairly direct proof that the second Borel cohomology group for $(\mathbb R, +)$ (with the trivial action on the circle group) is trivial? The motivation is to show ...
0
votes
0answers
61 views
“Reciprocal” of Schoenberg's theorem
Schoenberg's theorem states that for a (say, countable group) $G$ and any real valued conditionally negative type function $\psi$ on $G$, the function $e^{-t\psi}$ is positive definite, for any ...
0
votes
1answer
79 views
Schur's lemma for antiunitary operators on complex Hilbert spaces
Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that
...
5
votes
0answers
184 views
Unitary representations of Tarski Monsters and other beasts
Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...
6
votes
0answers
208 views
Does every commutative $*$-algebra of operators on a prehilbert space have a character?
My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...
4
votes
0answers
205 views
Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep
Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
7
votes
2answers
126 views
Can monomial representations induced from nonmonomial representations?
Let $H$ be a subgroup of $G$. Let $\rho$ be an irr representation of $G$ induced from an irr representation $\theta$ of $H$. It is well known that $\rho$ is monomial if $\theta$ is monomial. Is it ...
3
votes
1answer
239 views
Arthur's refinement of parameters for unitary automorphic representations
In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...
3
votes
0answers
134 views
Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
1
vote
1answer
28 views
Analytic criteria for the support of the Plancherel measure for SL(2,Qp), spherical functions
Is there an analytic criteria to determine the support of the Plancherel measure for SL(2,Qp). At least for unitary spherical representations
1
vote
0answers
108 views
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 $H_\pi$ ...
6
votes
2answers
147 views
Spherical functions for sl(2,Q_p)
I kindly would like to ask you the following- I am refering to page
175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms
..."
My question to which I would kindly ask you to answer ...
7
votes
0answers
293 views
Unique maximal ideal in group C*-algebras
Let $G$ be a discrete group. Let $C^*(G)$ denote the full group C*-algebra of $G.$ Let $\pi:C^*(G)\rightarrow \mathbb{C}$ be the *-homomorphism associated with the trivial representation of $G.$
...
1
vote
0answers
24 views
About Blattner`s generating function in the holomorphic case
If $(\pi_\lambda, H_\lambda)$ is a holomorphic discrete series with Harish-Chandra parameter $\lambda$, it is known that $H_\lambda$ decomposes as K-module as $V_\Lambda \otimes S(p^+)$ where ...
1
vote
0answers
70 views
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 ...
3
votes
0answers
108 views
Character of continuous series representation of GL(2)
It is wellknown that the character of an irreducible, unitary representation of $GL(n,\mathbb{C})$ uniquely determines the isomorphism classes. I fail to construct a function for $GL(2, \mathbb{C})$, ...
3
votes
1answer
136 views
Supercuspidal with Iwahori fixed vector
Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, ...
2
votes
0answers
58 views
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 ...
17
votes
2answers
406 views
Which groups are the unitary group of a $C^*$-algebra
Which groups are the unitary group of a $C^*$-algebra?
Does anyone know anything in this direction?
1
vote
1answer
139 views
Fell topology in terms of distributions
Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi_j \rightarrow \pi$ if and only if $tr\; \pi_j ...
6
votes
1answer
143 views
Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?
Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...
4
votes
0answers
212 views
How to find the unitary matrices in this exponential matrix representation
In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and ...
12
votes
3answers
602 views
Representing SU(3) with 3 ropes in 3 dimensions
The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...
1
vote
1answer
129 views
Embedding of Two Objects Into Higher Dimensions With Their Sum
Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...
8
votes
0answers
188 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
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0answers
121 views
Do local L-functions/epsilon factors vary continuously with the Fell topology?
Edit due to the comment.
Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.
Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
4
votes
1answer
287 views
How to translate the representation theory of semisimple to reductive groups?
I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the ...
7
votes
1answer
389 views
Quantum Cellular Automata on Riemannian manifolds and geometric group theory
We try to motivate our question. We have a certain logical/operational structure that has an
emergent physical interpretation. We are giving this structure a geometric setting via
quasi-isometries. ...
10
votes
3answers
428 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
209 views
Quantized conserved quantities appearing from the Lie-algebra
Hi,
consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to ...
5
votes
2answers
243 views
Steinberg reps of reductive groups over local fields vs finite fields
Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$.
Edit: The statements only make sense modulo tensoring by one-dimensional representations.
Are the unitary, ...
7
votes
1answer
639 views
Unbounded representations of groups
Let $H$ be a Hilbert space and $G$ be a finitely generated group. Let $\pi:G\rightarrow GL(H)$ be a representation.
A map $c:G\rightarrow H$ is called cocycle if $c(gh)=π(g)c(h)+c(g)$ for all $g,h$ ...
7
votes
1answer
362 views
Trace Class Functions on locally compact groups
Let $G$ be a locally compact subgroup, $\mu$ a Haar-measure.
For $f \in L^1(G)$, and for $\pi$ a unitary, topology irreducible, representation of $G$ on
an Hilbert space $H_\pi$, it is customary to ...
5
votes
1answer
370 views
Faithful representation of the projective unitary group with the lowest dimension?
What is the lowest dimension of a faithful ordinary representation (as compared with projective representation) of the projective unitary group $\rm{PU}(d)$? Is it $d^2-1$?
3
votes
2answers
213 views
Proper subgroups of $\rm{SU}(d)$ that act transitively on $\rm{CP}^{d-1}$?
The special unitary group $\rm{SU}(d)$ has a canonical action on the Hilbert space of dimension $d$, and this action induces a canonical action on the projective space $\rm{CP}^{d-1}$, which is ...
3
votes
1answer
163 views
Dense subspaces in primitive ideals of C-star algebras
Let $G$ be a unimodular locally compact group (my main examples are algebraic groups over local fields. Thefore we can assume $G$ is Type I, if necessary). Then there are at least three group algebras ...
2
votes
1answer
145 views
Unitary representations of a group given generating set
A group $G$ is generated by $1, -1, g_1, g_2, \ldots, g_n$. The relation of its generators is given by a simple undirected graph $G = (V=[n], E)$, where $(i, j) \in E$ means $g_i g_j = -g_j g_i$. In ...
15
votes
1answer
515 views
Is a reductive adelic group a Type I group?
I foresee that to experts of automorphic forms this question will sound unimportant or useless or even not worthy of an answer; but none of these are going to stop me from asking it!
The question is ...
0
votes
2answers
439 views
Similarity about unitary matrices
Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting, as well as $F_1, \ldots, F_k$. If they are similar, i.e., there exists $T \in GL_n(\mathbb{C})$ such that
$$
G_i = T^{-1} F_i T
...
6
votes
2answers
233 views
Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group
Hi All,
I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:
I am trying to understand the structure (e.g., decomposition) of the unitary ...
4
votes
2answers
257 views
comprehensive presentation of the unitary dual of $SO_0(n,1)$
The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case ...
13
votes
2answers
823 views
Regarding Cayley Graphs of Property (T) Groups
A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...
2
votes
1answer
170 views
Positive definite functions on G from Hilbert space vectors?
Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$?
This question is rather vague and ...
10
votes
1answer
422 views
Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am ...
7
votes
2answers
648 views
Induced representations of topological groups
Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from http://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups)
If $G$ is a group ...
7
votes
1answer
858 views
Unitary representations of the ax+b group: an accessible presentation
The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described ...
12
votes
3answers
1k views
Positive definite function zoo
I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a ...
3
votes
4answers
780 views
Finite-dimensional faithful representations of compact groups
Is it true that a compact group always has a faithful, finite-dimensional unitary representation? If not, are there any reasonably simple counter-examples?
I've done some research and know that every ...
