Automorphism group of flag manifolds?

Question

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 $A$?

Answer 1

Claudio's answer settles the determination of $\DeclareMathOperator\Aut{Aut}\Aut^{\mathrm{o}}(F)$; that of $\Aut(F)$ is more subtle. For Grassmannians this is a classical result of Chow (On the geometry of algebraic homogeneous spaces, Ann. of Math. (2) 50 (1949), 32-67): $\Aut(\mathbb{G}(p,n))$ is connected (hence equal to $\operatorname{GL}(n)$) except in the case $n=2p$, where $\pi _0(\Aut(\mathbb{G}(p,2p ))=\mathbb{Z}/2$. This has been extended to all flag varieties by Tango: On the automorphisms of flag manifolds. Bull. Kyoto Univ. Ed. Ser. B No. 49 (1976), 1–9. He proves that the automorphism group of $\mathbb{F}(n;n_1,\dotsc ,n_k)$ is connected except in the case $n_i+n_{k-i}=n$ for all $k$, where it has 2 components.

Answer 2

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSO{PSO}$Irreducible complex flag manifolds are of the form $X=G/P=G_u/K$ where $G$ is a complex simple Lie group and $P$ is a parabolic subgroup, and $G_u$ is a compact real form of $G$ and $K=P\cap G_u$. As a rule, the connnected automorphism group $\Aut^0(X)$ coincides with $G$, with the only exceptions:

1. $X=G_2/U_2$, $\Aut^0(X)=\SO_7(\mathbb C)$.
2. $X=\Sp_r/\Sp_{r-1}U_1$, $\Aut^0(X)=\PSL_{2r}(\mathbb C)$.
3. $X=\SO_{2r+1}/U_r$, $\Aut^0(X)=\PSO_{2r+2}(\mathbb C)$.

You can find a discussion in chapters 3 and 4 in Lie group actions in complex analysis, by Dmitri N. Akhiezer, Aspects of Mathematics, vol. E27, Friedr. Vieweg, Braunschweig and Wiesbaden, 1995.

Edit: Thanks to abx for the further explanations and original papers. I just wanted to point out that the reducible case and the full automorphism group are also discussed in the book by Akhiezer. In the reducible case, the connected automorphism group splits as the direct product of the connected automorphism groups of the factors. Moreover $\Aut(X)$ is the semidirect product of $\Aut^0(X)$ by a finite group isomorphic to the subgroup of the automorphism group of the Dynkin diagram of the complex semisimple Lie group $\Aut^0(X)$ preserving the vertices corresponding to the parabolic subalgebra.

Answer 3

While the question and answers have explained different aspects of the story for complex groups and homogeneous spaces, it might be worthwhile to place the question in a broader context. The complex groups here can just as well be regarded as algebraic groups, with the underlying field algebraically closed of any characteristic. The picture is remarkably similar in all cases, though expressed in somewhat different language. In any case, for any homogeneous space the starting point is the ambient group acting naturally, but there may be further automorphisms.

Two classical references of interest are available online (though in French). Both are rather short but also somewhat sophisticated:

Demazure treated flag varieties in the context of algebraic geometry here. As he remarks, the complex situation had already been handled by Tits here.

The varieties or manifolds of interest all have the form $G/P$ but can be studied from different angles. Aside from this, I've added a tag to emphasize the group-theoretic setting.

Answer 4

See the paper Self-maps of homogeneous spaces by Srinivas and Paranjape, Inventiones mathematicae, 1989, vol. 2. That paper addresses precisely this question.