computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit Let $G$ be a compact , connected and simply connected Lie group  and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ for some $n$. Is there any counter example? 
 A: No, there are no counter-examples. Note that a generic coadjoint orbit is $G$-equivariantly diffeomorphic to $G/T$, for a maximal torus $T\subseteq G$. However, $G/T$ (also known as the full flag variety of $G_{\mathbb{C}}$) has a cell decomposition into even-dimensional cells (the Bruhat decomposition). Therefore, $G/T$ has vanishing integral cohomology in odd degrees, and each even-dimensional integral cohomology group is a free $\mathbb{Z}$-module. In particular, $H^2(G/T;\mathbb{Z})$ is a free $\mathbb{Z}$-module, so your statement is true in general.
A: The cohomology of regular coadjoint orbits of a compact Lie group can be computed using Morse theory.  This approach is due to Bott and Samelson (1955).  Let $A \in \mathfrak{g}^{*}$ be a regular element, $\langle\cdot,\cdot\rangle$ the Cartan-Killing form, and consider the coadjoint orbit $\mathcal{O}_{B}$ of a regular element $B \in \mathfrak{g}^{*}$, then $\langle A,\cdot\rangle: \mathcal{O}_{B}\to\mathbb{R}$ is a Morse function whose critical set is the Weyl orbit of $B$, and the index of a critical point $w(B)$ turns out to be twice the number of root hyperplanes crossed by traversing the line from $A$ to $w(B)$.  This gives a cell decomposition concentrated in even degrees, so the cohomology vanishes in odd degrees and in particular $H^{2}(\mathcal{O}_{B},\mathbb{Z})$ has the form you suggest.
