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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 $...
Michał Oszmaniec's user avatar
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 ...
Saal Hardali's user avatar
  • 7,789
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 ...
Cameron Zwarich's user avatar
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 ...
Peter Kravchuk's user avatar
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 ...
Tim Phalange's user avatar
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 ...
Transcendental's user avatar
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 ...
Giuseppe Negro's user avatar
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 $...
geometricK's user avatar
  • 1,903
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 ...
ThiKu's user avatar
  • 10.4k
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 ...
Michael_1812's user avatar
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 $V^\omega$ ...
Amin's user avatar
  • 399
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 ...
Michael_1812's user avatar
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 ...
A beginner mathmatician's user avatar
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 ...
Michał Oszmaniec's user avatar
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/...
Jeep Wrangler's user avatar
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 ...
John Jiang's user avatar
  • 4,466
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 ...
user avatar
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?
Sergii Voloshyn's user avatar