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?



Is $G/T$ a projective variety?
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 $\mathfrak{g}\_{\mathbb{C}}$.


