If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.

In the simplest case of $\mathbb{CP}^n$ this is just $PGL_{n+1}(\mathbb{C})$, though the proof relies on knowing that the Picard group of $\mathbb{CP}^n$ is $\mathbb{Z}$. I'm not even sure what to expect for the next-simplest case of Grassmannians. There's always a sizeable action of a general linear group, but I don't know if it's onto.

Sorry if this is well-known but I haven't been able to find the answer anywhere. If it's so well-known that I should be ashamed, then I'd also ask: is it known for flags of type other than $A$?