Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$ where $P$ here is Parabolic subgroup. Is there any method or referrence ?
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1$\begingroup$ Why do you call $\pi_2$ the "fundamental group"? $\endgroup$– Jason StarrCommented Feb 6, 2014 at 20:08
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$\begingroup$ You use the homotopy exact sequence, and you know that $G/P$ is connected and simply connected. You know the fundamental group of $G$ (see Serre, Complex Semisimple Lie Algebras). That should get you started, but I can't remember if I ever worked out the answer to this question. $\endgroup$– Ben McKayCommented Feb 6, 2014 at 20:18
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1$\begingroup$ There is a plenty of material on cohomology of generalizaed flag varieties. The cohomology of generalized flag varieties with coefficients in $\mathbb{C}$ is basically computed by Kostant's theorem from 62. $\endgroup$– Vít TučekCommented Feb 6, 2014 at 21:54
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2$\begingroup$ en.wikipedia.org/wiki/Bruhat_decomposition $H_2 = \mathbb Z^{rank(G)-rank(P)}$ This is not a research-level question. $\endgroup$– Allen KnutsonCommented Feb 7, 2014 at 3:21
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1$\begingroup$ Yes, as is directly derived from the Bruhat decomposition of $G^{\mathbb C}/P$. $\endgroup$– Allen KnutsonCommented Feb 8, 2014 at 5:36
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1 Answer
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As @abx has indicated, $\pi_2(G_{\mathbb{C}}/P)$ is isomorphic to $H_2(G_{\mathbb{C}}/P)$. To describe the latter group, we use the Bruhat decomposition of $G_{\mathbb{C}}/P$. Choose a maximal torus $T$ and Borel $B$ satisfying $T\subseteq B\subseteq P$. We then have the Bruhat decomposition $$G_{\mathbb{C}}/P=\coprod_{[w]\in W/W_P}BwP/P,$$ where $W_P$ is the subgroup of $W$ generated by the reflections for the simple roots whose negative root spaces belong to the Lie algebra of $P$. The dimension of the Bruhat cell indexed by a coset in $W/W_P$ is twice the length of the coset representative of minimal length. So, all cells are even-dimensional, and $H_2(G_{\mathbb{C}}/P)$ is the free abelian group of rank equal to the number of cosets in $W/W_P$ with minimal representatives of length $1$. So, this rank is equal to the number of simple roots whose negative root spaces are not in the Lie algebra of $P$. (If $P=B$, then this rank is precisely the number of simple roots.) I hope this is the sort of answer you are seeking.
answered Feb 6, 2014 at 22:01