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I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology …

Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{ …

Let $G$ be an even-dimensional compact Lie group with Lie algebra $\mathfrak{g}$ and let $T \subset G$ be a maximal torus with Lie algebra $\mathfrak{t}$. We can construct a left-invariant complex st …

Let $\mathfrak g$ be the Lie algebra of a compact connected Lie group $G$. Let $\mathfrak g_{\mathbb{C}}$ be the complexification of $\mathfrak g$ and let $\mathfrak h \subset \mathfrak g_{\mathbb{C}} …

Let $G$ be a compact Lie group having a left-invariant complex structure $J$. Is there a hermitian metric $h$ in $G$, compatible with the complex structure $J$, such that $G$ is a Kähler manifold? I …

On Helgason's book "Differential Geometry, Lie Groups, and Symmetric Spaces" it is said that the Lie derivative along a left-invariant vector field of an harmonic form is again a harmonic form. This a …

Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with …