I am interested in understanding the Picard group of orthogonal Grassmanians in the following two cases. If someone could give me the proofs or a hint or a good reference it would make me happy.
Case 1: Let $V$ be an even dimensional vector space say of dimension $n=2m$, let $k\leq m-2$ and let $Q\subset \mathbb{P}(V)$ be a quadric which is a simple cone (by which I mean that the corresponding symmetric bilinear form has corank 1). Then I believe the Picard group of $OGr(k,V)=\{U\subset Gr(k,V)|\mathbb{P}(U)\subset Q\}$ is $\mathbb{Z}$ and a generator is obtained by pulling back a generator for $Pic(\mathbb{P}^N)$ via the Plücker embedding.
Case 2: Let $V$ be a odd dimensional vector space of say dimension $n=2m-1$. Let $Q\subset \mathbb{P}(V)$ be a smooth quadric hypersurface. Let $k\leq m-2$. Then again I believe that the Picard group of $OGr(k,V)=\{U\subset Gr(k,V)|\mathbb{P}(U)\subset Q\}$ is $\mathbb{Z}$ and a generator is obtained by pulling back a generator for $Pic(\mathbb{P}^N)$ via the Plücker embedding.