Questions tagged [coadjoint-orbit]
Given a Lie group $G$, it acts smoothly on the dual $\mathfrak g^*$ of its Lie algebra $\mathfrak g$ by the coadjoint action. The orbits of that action are called coadjoint orbits.
15 questions
9
votes
1
answer
157
views
Eigenfunctions of the Laplace–Beltrami operator on the coadjoint orbit of $\mathfrak{su}(n)$
$\DeclareMathOperator\SU{SU}$For $\mathfrak{su}(2,\mathbb{C})$, the generic coadjoint orbit is $\mathbb{S}^2$, and the Laplace–Beltrami operator on it is given by
$$
\Delta \equiv \frac{1}{\sin\theta} ...
- 195
3
votes
0
answers
124
views
Branching problem of representation of Lie groups and orbit method
Branching problem asks how a restriction of an irreducible representation of $G$ to a subgroup $H$ decomposes. In case of (real) Lie groups, after labeling irreducible representations via highest ...
- 2,215
1
vote
0
answers
70
views
Minimal $K$-orbit on $\mathfrak{g}$
Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
- 951
1
vote
1
answer
340
views
Problem in understanding the coadjoint action of $\mathfrak {g}^{\ast}$ on $\mathfrak {g}$
$\DeclareMathOperator\ad{ad}$Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}$. Let the brackets on $\mathfrak {g}$ and $\...
- 199
5
votes
1
answer
203
views
Can all hermitian symmetric spaces be realised as coadjoint orbits?
Here is what I know. Assume $M\cong G/K$ is an irreducible hermitian symmetric space. Denote the Lie-algebra of $K$ by $\mathfrak{t}$. Proposition 1.2. chapter 3 in
Wienhard - Bounded cohomology and ...
- 61
0
votes
0
answers
150
views
Compact coadjoint orbits
The following statement is from the article Compact Coadjoint Orbits by John Rawnsley:
If $\mathcal{O}$ is a compact coadjoint orbit for the group $G$ then there is a closed normal subgroup $H$ of $G$...
- 139
2
votes
2
answers
290
views
Question about the Kähler structure on generic coadjoint orbits
Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}_r: = G\cdot r$ be a generic ...
- 139
5
votes
1
answer
187
views
Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.
One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan ...
- 5,701
1
vote
0
answers
69
views
In a contact Lie algebra, when is the Reeb vector a semisimple element?
Below is a question I have come across in my research, and it seems like a question that has been answered (or at least asked) in the past; however, I have been unable to find any references that ...
- 11
9
votes
2
answers
1k
views
Meaning of the coadjoint representation and its orbits
Given a Lie group $G$ there is a natural representation of $G$ on the dual of its Lie algebra $\mathfrak{g}^*$ given by the coadjoint representation. This representation is obtained by differentiating ...
- 1,474
6
votes
1
answer
369
views
Can I bring the Kirillov 2-form on coadjoint orbits to adjoint orbits?
I tried asking this question on stackexchange and received no response.
Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I ...
user125834
4
votes
0
answers
139
views
"Signature Changing" Generalization of Lie Algebra?
I have in mind a mathematical structure I've never heard of before. Does anyone know what might be?
It is a manifold with vector fields whose Lie brackets have structure coefficients that are ...
2
votes
1
answer
137
views
Projections of orbifolds
A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an ...
- 23
2
votes
0
answers
260
views
A representation similar to coadjoint representation?
In a project of quantization, I come up with a finite dimensional representation of $so(d)$ that I wish to find some decent references for it. I guess it could have been studied thoroughly in ...
- 129
2
votes
1
answer
285
views
The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$
Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let $...
user21574