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0
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1answer
66 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 ...
4
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0answers
162 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
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0answers
183 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
191 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
123 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
214 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
131 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
27 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
105 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
137 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
279 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
23 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
67 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
102 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
126 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
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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
397 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
123 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
133 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
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0answers
184 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
594 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
125 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
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0answers
182 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
108 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
272 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
379 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
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3answers
391 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
194 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
180 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
633 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
346 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
358 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
203 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
161 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
144 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
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1answer
478 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
423 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
228 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
253 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
813 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
159 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
394 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
640 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
810 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
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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
762 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 ...
4
votes
3answers
813 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
176 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., ...
7
votes
2answers
577 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
507 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 ...