All Questions
18 questions
9
votes
1
answer
414
views
Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces
If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by ...
CommunityBot
- 1
1
vote
1
answer
423
views
Quaternion representation and Haar measure of $SU(3)$ [closed]
Do we have easily and practically useful quaternion representation for $SU(3)$ group element and for Haar measure?
Also, is $SU(2)$ really simplified in the quaternion base?
- 188k
3
votes
1
answer
451
views
Topological vector spaces in direct sum
A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...
- 603
3
votes
0
answers
212
views
Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory
Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$:
$$
A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~,
$$
and let ${\mathbb{V}}$ be decomposable into a ...
- 603
3
votes
0
answers
269
views
Kazhdan Property T of semisimple Lie groups
I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M.,
Analogs of Wiener's ergodic theorems for semisimple Lie groups. II.
Duke Math. J. 103 (2000), no. 2, 233–259] (MSN).
I want to ...
- 12.9k
7
votes
0
answers
420
views
What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?
Here is the story as I see it.
Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
- 557
4
votes
2
answers
313
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
- 692
4
votes
0
answers
279
views
Cyclic vectors for regular representations
I'm looking for references about the following aspect of cyclic vectors for regular representations.
Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $...
- 31.5k
14
votes
1
answer
514
views
Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?
Motivating examples:
Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$
The ...
CommunityBot
- 1
8
votes
1
answer
584
views
Tensor products of unitary irreducible representations of $SU(2,2)$
What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...
- 31.5k
4
votes
0
answers
315
views
Compactly supported distributions as a projective G-module
For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
- 10.4k
2
votes
1
answer
529
views
Cotangent bundle of symmetric space is symmetric space?
Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...
- 356
5
votes
1
answer
459
views
Some questions on analytic vectors and the integrability of Lie-algebra representations
I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
- 1,396
3
votes
1
answer
310
views
Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?
Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
3
votes
2
answers
236
views
Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...
- 10.3k
2
votes
1
answer
544
views
Characters separating points on Maximal Torus modulo Weyl group?
Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group.
Every finite-dimensional representation of G has a character, which is a function on G, T and T/...
- 52.9k
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
2
votes
2
answers
551
views
L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 23k