What is the Picard group of a variety of type G/P with $G$ a semi simple algebraic group and $P$ a parabolic subgroup?

Question

Let $G$ be a simple algebraic group and let $P$ be a parabolic subgroup of $G$. It follows $X:=G/P$ is a smooth projective variety - the flag variety of $G$ corresponding to $P$.

Is it true that the following holds:

Pic($X$) has rank $1$ iff $P$ is a maximal parabolic subgroup.

Why? Where do i find a reference?

Answer 1

I'm not really sure what you're asking here--a maximal parabolic subgroup of $G$ is $G$ itself! That said, I'm fairly sure the below will answer your question.

Note that if one has a parabolic subgroup $P$ of $G$, a split connected group, then one has an exact sequence

$$0\to X^\ast(G)\to X^\ast(P)\to \mathrm{Pic}(G/P)\to \mathrm{Pic}(G)\to\mathrm{Pic}(P)\to 0$$

(e.g. see [Mil, §18.f]). From this, we see that if $G$ is semisimple then $X^\ast(G)=0$ and $\mathrm{Pic}(G)$ is finite (e.g. see [Mil, Corollary 18.23]) and thus this above says that we have an injection $X^\ast(P)\to \mathrm{Pic}(G/P)$ with finite quotient. Thus, the rank of $X^\ast(P)$ and $\mathrm{Pic}(G/P)$ are the same. But, since any map $P\to\mathbb{G}_m$ factors through $P/R_u(P)$ if $L$ is a Levi factor of $P$ we see that $X^\ast(P)=X^\ast(L)$. Moreover, we see that any map $L\to\mathbb{G}_m$ factors through $L^\mathrm{ab}$ and, in fact, the rank of $X^\ast(L)$ is the dimension of $L^\mathrm{ab}$ which is equal to the dimension of $Z(L)$ by standard theory (e.g. see [Mil, Example 19.25]).

Putting all this together we deduce the following:

Fact: Let $P$ be a parabolic subgroup of a semisimple group $G$. Then, the rank of $\mathrm{Pic}(G/P)$ is equal to $\dim L^\mathrm{ab}=\dim Z(L)$ where $L$ is a Levi factor of $P$.

Answer 2

I do not have a precise reference for this. Let $G:=SL(V)$ with $V\cong k\{e_1,..,e_d\}$ with $k$ a field. Given any parabolic subgrop $P$ of $SL(V)$, there is a flag $V_{\bullet} \subseteq V$ such that $P$ is the subgroup of elements stabilizing the flag: Let

$$ 0 \neq V_1 \subsetneq V_2 \subsetneq \cdots \subsetneq V_{l} \subsetneq V $$

be such a flag $V_{\bullet}$ in $V$ with $dim(V_i):=d_i$. Let $P \subseteq SL(V)$ be the (parabolic) subgroup fixing the flag $V_{\bullet}$. It follows $SL(V)/P$ is the "flag variety" parametrizing flags of type $V_{\bullet}$ in $V$. I believe there is an isomorphism

$$ u:\mathbb{Z}^l \cong \operatorname{Pic}(SL(V)/P) $$

induced by the closed embedding

$$ \phi: SL(V)/P \rightarrow \mathbb{G}(d_1,V) \times \cdots \times \mathbb{G}(d_l,V)$$

defined by

$$\phi([V_{\bullet}]):=([V_1],..., [V_l]).$$

There is an isomorphism

$$\operatorname{Pic}(\mathbb{G}(d_i,V))\cong \mathbb{Z}$$

with generator $\mathcal{O}(1)$. Hence if

$$p_i: \mathbb{G}(d_1,V) \times \cdots \times \mathbb{G}(d_l,V)\rightarrow \mathbb{G}(d_i,V)$$

is the projection, it follows any invertible sheaf $\mathcal{L}\in \operatorname{Pic}(SL(V)/P)$ is on the form

$$\mathcal{L}\cong \phi^*(p_1^*\mathcal{O}(n_1)\otimes \cdots \otimes p_l^*\mathcal{O}(n_l)).$$

Hence the isomorphism $u$ is the following map:

$$u(n_1,..,n_l):= \phi^*(p_1^*\mathcal{O}(n_1)\otimes \cdots \otimes p_l^*\mathcal{O}(n_l)).$$

Question: "$Pic(X)$ has rank $1$ iff $P$ is a maximal parabolic subgroup. Why? Where do i find a reference?"

Anser: The answer is "no" for the following reason:

The quotient $\operatorname{Pic}(SL(V)/P)\cong \mathbb{Z}$ iff $l=1$ which is iff $SL(V)/P\cong \mathbb{G}(d_1,V)$ is the grassmannian of $d_1$-planes in $V$.

Over the complex numbers I believe you may find this in:

Akhiezer, Dmitri N. "Lie group actions in complex analysis". Aspects of Mathematics. E27. Braunschweig: Vieweg. vii, 201 p. (1995).

In particular page 65 in the above book mentions the following: The quotient $SL(V)/P$ is projective iff $P$ is parabolic. Moreover for any parabolic subgroup $P$, there is a flag $V_{\bullet}$ in $V$ with the property that $P$ is the subgroup stabilizing the flag.

If $G$ is an "affine algebraic group" (of finite type) over a field $k$ you may always realize $G$ as a closed subgroup of $GL_k(V)$, where $V$ is a finite dimensional $k$-vector space, and if $G$ is simple (or semi simple) you may ask for similar results: "Is every parabolic subgroup of $G$ a subgroup stabilizing a flag in $V$?" and "Is the quotient $G/P$ projective iff $P$ is parabolic?". I believe the answers to these questions is "well known" but I do not have a precise reference.