1
vote
0answers
123 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 ...
3
votes
1answer
202 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 ...
4
votes
0answers
75 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
1
vote
0answers
74 views

Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...
4
votes
1answer
462 views

A question on Grassmannian

Let $V$ be the space of $4$ by $4$ Hermitian matrices, that is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform measure of $$ \left\{ W\in Gr\left(5,V\right):W \text{contains no ...
4
votes
0answers
264 views

Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following: Is it true that all ...
3
votes
2answers
165 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 $V^\omega$ ...
2
votes
1answer
244 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 ...
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 ...
2
votes
0answers
180 views

distribution on Lie Groups and representations

Let $G$ be a Lie group and $\pi$ a continuous action on $V$ a Fréchet space. This action induces a representation of the compact-supported function $C_c(G)$ (with convolution as product) by $f\in ...
3
votes
3answers
373 views

are the smooth vectors of a Frechet space dense?

Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
17
votes
3answers
809 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 ...
2
votes
2answers
283 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 ...
14
votes
4answers
1k views

Representing a product of matrix exponentials as the exponential of a sum

In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
7
votes
1answer
464 views

Is $SU(\infty)$ amenable?

We can write the finitary special unitary group $SU(\infty)$ as the direct limit $\varinjlim SU(n)$ of ordinary special unitary groups. These groups $SU(n)$ are compact, thus amenable. In other ...
7
votes
0answers
314 views

For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?

The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
11
votes
3answers
613 views

Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...