Picard group of a quotient of a group by its maximal parabolic subgroup

Question

Let $G$ be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero and $P$ be the maximal parabolic subgroup. We know that the quotient $Z=G/P$ is a projective scheme. Is it true that the Picard group of $Z$ is isomorphic to $\mathbb{Z}$?

Answer 1

Yes. We can assume that $G$ is simply connected, and in this case, any line bundle on $G/P$ is equivariant for $G$ (if $G$ isn't simply connected, you get silly things like $\mathcal{O}(1)$ on $\mathbb{P}^n$ isn't equivariant for $PSL_{n+1}$). For any maximal parabolic subgroup, $P/[P,P]\cong \mathbb{G}_m$, and every line bundle is gotten by taking $(G\times V)/P$ for $V$ a 1-dimensional representation of $P$.