0
votes
0answers
78 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 ...
1
vote
0answers
22 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 ...
3
votes
0answers
100 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 $GL(2, \mathbb{C})$, ...
2
votes
0answers
56 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 ...
1
vote
1answer
105 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
125 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 ...
12
votes
3answers
590 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 ...
8
votes
0answers
179 views

when do norm-continuous unitary representations separate points of a group?

Recently I found in the web a discussion on the following question: ...
4
votes
1answer
263 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 ...
10
votes
3answers
350 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 ...
4
votes
1answer
176 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
628 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
334 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
279 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 $d^2-1$?
4
votes
2answers
246 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
786 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
155 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
362 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 ...
3
votes
4answers
738 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 ...
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., ...
7
votes
2answers
568 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 ...
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
599 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
434 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 ...
15
votes
4answers
3k 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 ...
37
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 ...
3
votes
1answer
285 views

What is the “right” hermitian structure on tensor products of quantum group representations?

This is pretty specific, but there are some experts around. So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...