Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My question is how can we find a formula for $\pi_1(G_a)$ by using maximal tours. Is there any relation?
1 Answer
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Assume that $G$ is connected and simply-connected. From the principal $G_a$-bundle $G_a\rightarrow G\rightarrow G/G_a=\mathcal{O}_a$, we obtain a long-exact sequence $$\ldots\rightarrow\pi_i(G_a)\rightarrow\pi_i(G)\rightarrow\pi_i(\mathcal{O}_a)\rightarrow\pi_{i-1}(G_a)\rightarrow\ldots.$$ Since $\pi_1(G)=0$ and $\pi_2(G)=0$, exactness considerations imply that $\pi_1(G_a)=\pi_2(\mathcal{O}_a)$. Often, orbits $\mathcal{O}_a$ are partial flag varieties of $G_{\mathbb{C}}$, and their second homotopy groups are reasonably well-known. I am not sure if this helps.
answered Feb 6, 2014 at 19:50
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$\begingroup$ Peter , The fact is I am trying to find $\pi_2(\mathcal{O}_a)$. Thats why I posted this question. $\endgroup$– user21574Commented Feb 6, 2014 at 19:53
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$\begingroup$ Look for $\pi_2$ of partial flag varieties $G_{\mathbb{C}}/P$, where $P\subseteq G$ is a parabolic subgroup. $\endgroup$ Commented Feb 6, 2014 at 19:54
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$\begingroup$ Do you have any referrence to finding the $\pi_2$ of generalized flag variety ? $\endgroup$– user21574Commented Feb 6, 2014 at 19:56