TITLE: Canonical bundle of the Lagrangian Grassmannian
QUESTION [5 upvotes]: I work through the paper On branched coverings of some homogeneous space of Kim and Manivel and I came across the definition of the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of $n$-dimensional lagrangian subspaces of a $2n$-dimensional symplectic vector space. In the paper it is called 'a fact' that the canonical bundle $K_{\mathbb{LG}_n} = \mathcal{O}_{\mathbb{LG}_n}(-n-1)$, but I'm not able to verify it. Can someone help me in this?
Thanks in advance.

REPLY [10 votes]: Here is a group theoretical solution which makes it is easy to compute the canonical bundle of any generalized flag variety.
The Lagrangian Grassmannian is the homogeneous space $G/P$ where $G=Sp(2n)$ and $P$ is the maximal parabolic subgroup corresponding to the set of simple roots $\{\alpha_1,\ldots,\alpha_{n-1}\}$. Then $Pic(G/P)$ equals $\Xi(P)$, the character group of $P$. The latter is cyclic and its generator $\eta:=\sum_{i=1}^n\epsilon_i$ corresponds to $\mathcal O(-1)$. The canonical bundle corresponds to the determinant character of $(\mathfrak g/\mathfrak p)^*\cong\mathfrak p_u$ and therefore to the sum of roots in $\mathfrak p_u$:
$$
\sum_{i<j}(\epsilon_i+\epsilon_j)+\sum_i(2\epsilon_i)=(n+1)\eta.
$$
Thus $K=\mathcal O(-n-1)$.