Is $G/T$ a projective variety? Let $G$ be  a  semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?
 A: Suppose that $G$ is compact, connected, and semisimple. Let $T\subseteq G$ be a maximal torus.  Take the complexification $G_{\mathbb{C}}$ of $G$, and choose a Borel subgroup $B\subseteq G_{\mathbb{C}}$ containing $T$. Using the Iwasawa decomposition $G_{\mathbb{C}}=GB$, we see that $G$ acts transitively on $G_{\mathbb{C}}/B$. Also, the stabilizer of the identity coset is $T$, giving us a $G$-equivariant diffeomorphism $G/T\cong G_{\mathbb{C}}/B$. 
The thing to note is that $G_{\mathbb{C}}/B$ naturally carries the structure of a complex projective variety. Hence, $G/T$ inherits the projective variety structure for which the isomorphism $G/T\cong G_{\mathbb{C}}/B$ is an isomorphism of projective varieties. Note, however, that the projective variety structure of $G/T$ depends on the choice of $B$ containing $T$, or equivalently, the choice of positive roots for the adjoint representation of $T$ on $\mathfrak{g}_{\mathbb{C}}$.
A: Try $G=SL(2,\mathbb{R})$; the $T$ is the stabilizer of a metric, so $G/T$ is the set of metrics on $\mathbb{R}^2$ with unit volume, certainly not projective, because they are just positive definite symmetric $2 \times 2$ matrices with determinant 1, an affine hypersurface. Projective varieties are compact.
