I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a complex vector space together with fixed symplectic bilinear form. I already know, that this is $K_{\mathbb{LG}_n} = \mathcal{O}_{\mathbb{LG}_n}(-n-1)$.
So far I know, that the tangent space in one point $L$ (that is a Lagrangian subspace) of $\mathbb{LG}_n$ can be identified with the set of symmetric bilinear forms $B_{symm}(L,L)$, see Piccione, Tausk - A Student's Guide to Symplectic Spaces.
For some references or help for the proof I'd be grateful.