# Tagged Questions

**0**

votes

**1**answer

100 views

### Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$.
Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k) $ be the complexified Lie group of $Z(k)$ ...

**0**

votes

**0**answers

283 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**4**

votes

**1**answer

101 views

### Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...

**8**

votes

**4**answers

286 views

### Automorphism group of flag manifolds?

If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.
...

**5**

votes

**1**answer

308 views

### Root space decomposition

What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...

**1**

vote

**1**answer

129 views

### the group of all biholomorphic group automorphisms of complex tori

My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let ...

**2**

votes

**2**answers

181 views

### compute the automorphism of Iwasawa manifold

An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup.
We can also refer to Griffiths and Harris's Principles of Algebraic Geometry ...

**5**

votes

**1**answer

381 views

### Geometric structure of flag manifolds, Borel -Weil-Bott theorem

I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be.
Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a ...

**10**

votes

**3**answers

414 views

### orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily ...

**2**

votes

**1**answer

147 views

### Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain

Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?

**1**

vote

**1**answer

79 views

### $SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...

**22**

votes

**3**answers

831 views

### Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...

**5**

votes

**0**answers

199 views

### Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...

**6**

votes

**1**answer

253 views

### Kähler form on complex Lie group

Hallo,
Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a ...

**2**

votes

**1**answer

197 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 ...

**1**

vote

**1**answer

291 views

### algebraic closure of Lie groups in

Let $G$ be a connected, simply connected, solvable, complex Lie group with a discrete subgroup $\Gamma$.
Let also $G_a$ be Hochshild-Mostow hull of $G$, i.e., there exists a solvable linear algebraic ...

**2**

votes

**2**answers

383 views

### Non-Integrable Almost-Complex Structures for Homogeneous Spaces

Let $M$ be a smooth homogeneous $G$-space for a Lie group $G$, and let $J$ be a $G$-invariant almost-complex structure for $M$. Do there exist succinct sufficient (and neccessary) conditions for $J$ ...

**1**

vote

**1**answer

199 views

### Does the diffeomorphism group preserving a particular section act transitively?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$, be the space of homogeneous
degree $d$ polynomials in three variables $[X,Y,Z] \in \mathbb{P}^2$ upto scaling, where
$\delta_d = \frac{d(d+3)}{2}$. ...

**9**

votes

**3**answers

831 views

### Complex structure on flag manifolds

Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. Then the flag manifold $G/T$ is a complex manifold and a symplectic manifold. One way to see the symplectic structure is to view $G/T$ ...

**4**

votes

**2**answers

398 views

### Semi-Simple Kahler Groups?

We say that a Kahler manifold is a Kahler group if it is also a Lie group. I would like to know which semi-simple Lie groups are also Kahler groups?

**4**

votes

**2**answers

383 views

### Equivariance of vector bundles over G/B

Let $G$ be a complex semisimple group, $B$ a Borel subgroup of $G$ and $X=G/B$ the flag variety of $G$. If $G$ is simply connected, then every line bundle $L$ on $X$ can be made $G$-equivariant (see ...

**3**

votes

**1**answer

345 views

### Dolbeault Operators for $CP^1$ as $\mathfrak{su}(2)$ Actions.

This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.
Put a grading on the ...

**3**

votes

**2**answers

582 views

### Why can the Dolbeault Operators be Realised as Lie Algebra Actions

I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that ...