The unitary-representations tag has no wiki summary.
5
votes
0answers
105 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
20 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
61 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
90 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
96 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
54 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 ...
4
votes
0answers
167 views
An example of group with specific properties of its action on a discrete set
I am looking for an example of a discrete group $G$ which satisfies the following conditions:
$G$ acts on a set $X$ transitively and has amenable stabilizers.
There are finite subsets of ...
18
votes
2answers
355 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
94 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
104 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 ...
5
votes
0answers
151 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
581 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 ...
2
votes
1answer
123 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 ...
9
votes
0answers
175 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
0answers
91 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
252 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 ...
6
votes
1answer
351 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
290 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
166 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 ...
4
votes
1answer
172 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, ...
8
votes
1answer
624 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
312 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
270 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
187 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
157 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
143 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 ...
13
votes
1answer
443 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
380 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
218 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
242 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
736 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
150 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 ...
9
votes
1answer
334 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
613 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
714 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 ...
11
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 ...
2
votes
4answers
713 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 ...
3
votes
3answers
730 views
What is the difference between a primary representation and a irreducible representation?
I am currently reading some of Mackey's work on unitary representation.
Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the ...
2
votes
1answer
175 views
Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., ...
6
votes
2answers
552 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 ...
3
votes
2answers
485 views
Does every nontrivial group adimit a nontrivial unitary representation?
For a finitely presented group, does there always exist a nontrivial finite dimensional unitary representation?
If two finitely presented groups have the same set of finite dimensional unitary ...
2
votes
1answer
158 views
Is there an abstract characterization of freeness in terms of additive unitary cocycles?
This question is very closely related to my other question here.
Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map ...
4
votes
2answers
2k views
Representations of Lorentz group
Questions:
What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
Why should one read Bargmann's paper on ...
2
votes
3answers
591 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 ...
3
votes
2answers
417 views
decomposition into irreducible unitary representations: references for explicit formulas?
I'm looking for references of the decomposition of $L^2(\Gamma\backslash G)$, where $G$ is a connected Lie group, and $\Gamma\subset G$ a discrete lattice; for simplicity one may assume that $G$ is ...
0
votes
0answers
233 views
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 ...
13
votes
3answers
485 views
Is there a characterization of free groups in terms of the unitary dual?
If $G$ is a countable discrete group, I'm curious if it is possible to decide whether $G$ is a free group only by looking at properties of $Rep(G)$, the collection of (equivalence classes of) strongly ...
14
votes
4answers
2k views
Unitary representations of SL(2, R)
I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being ...
23
votes
4answers
2k views
Induction and Coinduction of Representations
I'd like to understand the general framework of induction and coinduction of representations. If G is a finite group and H a subgroup, I know that there is a restriction functor from representations ...
35
votes
9answers
3k views
Is every finite group a group of “symmetries”?
I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually ...
