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I don't know whether I really get you point. The importance of $\pi_i G$ for higher $i$'s is that they determine whether a $G$-principal bundle can be trivialized. Remember a $G$-principal bundle $E\ …

In general cases Verma modules are corresponding to D-modules on flag variety and then via Riemann-Hilbert correspondence to constructible sheaves on flag variety. Hence the suitable generalization of …

There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by: $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw]. $$ Let $\mathbb{Z}/2=\{ 1,s \ …

Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that $\pi(g\cdot m) …

For your question 2, the reason is in fact we can prove $$ K\times \mathfrak{p}\xrightarrow{\sim} G\\ (k,p)\mapsto k\exp(p) $$ is an diffeomorphism. Here $\mathfrak{p}$ is the $-1$ eigen space of the …

Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$). My questions is: it is always true that we have …

Let $G$ be a compact connected semisimple Lie group, $\mathfrak{g}$ be its complexified Lie algebra and $\mathfrak{g}^*$ its complex dual space. We can form the symmetric algebra $S(\mathfrak{g}^ * ) …

This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We kn …

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h} …

For a first introduction you can read Michel Brion's "http://arxiv.org/pdf/math/0410240v1.pdf". He gives a nice introduction (for G=GL(n)) in Section 1. I'm not sure whether your curve method works b …

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 $\mathf …

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 t …