All Questions
10 questions
7
votes
0
answers
167
views
How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,019
6
votes
1
answer
674
views
Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...
- 3,399
5
votes
1
answer
902
views
Explicit description of the Lagrangian Grassmannian as a homogeneous space
Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit ...
- 459
3
votes
1
answer
284
views
Schubert cells in G/P for reductive G
All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
- 1,677
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
3
votes
1
answer
138
views
Euler characteristic of a holomorphic homogeneous vector bundle
Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...
- 643
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
1
answer
256
views
The real group orbits on the flag variety always contains the holomorphic directions?
Let $G$ be a real semisimple Lie group and $\mathfrak{g}$ be its complexified Lie algebra. We have the flag variety $\mathcal{B}$ of $\mathfrak{g}$ which is the set of all Borel subalgebras of $\...
- 9,019
1
vote
1
answer
408
views
De Rham cohomology of homogeneous spaces
Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
0
votes
2
answers
446
views
Classifying compact homogeneous Kähler manifolds
In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...
- 163