Tangent bundle for orthogonal and isotropic Grassmannians

Question

We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as $$ \mathbb G(k,V):=\{W \subset V : \dim W=k\}. $$ Then consider a non-degenerate quadratic form on $V$, we write $q: V \times V \to \mathbb C$ if it is symmetric, $\omega: V \times V \to \mathbb C$ if it is skew-symmetric (in this case $n$ is even). Then one can define, respectively, the orthogonal $k$-th Grassmannian $$ \mathbb O\mathbb G(k, V)=\{q\text{-isotropic }W \subset V : \dim W=k\} $$ and the isotropic $k$-th Grassmannian $$ \mathbb I\mathbb G(k, V)=\{\omega\text{-isotropic }W \subset V : \dim W=k\}. $$

In the "classic" case, we know that the tangent bundle is given by $$ T_{\mathbb G(k,V)}=\mathcal S^\vee \otimes \mathcal Q $$ where $\mathcal S$ and $\mathcal Q$ denote the universal subbundle and the universal quotient bundle, following Eisenbud and Harris. When $\dim V=2m$, then a similar expression is known for $\mathbb O \mathbb G(m,V)$ and $\mathbb I \mathbb G(m,V)$: $$ T_{\mathbb O \mathbb G(m,V)}=\wedge^2 \mathcal U^\vee, \quad T_{\mathbb I \mathbb G(m,V)}=S^2 \mathcal U^\vee $$ where $\mathcal U$ is given by the restriction of the tautological bundle on $\mathbb G(n,V)$.

Finally the question: are there similar expressions of the tangent bundles for the other orthogonal and isotropic Grassmannians? Also a reference would be enough.

Answer 1

The tangent bundle to the orthogonal Grassmannian fits into an exact sequence $$ 0 \to T_{\mathrm{OG}(k,V)} \to \mathcal{S}^\vee \otimes \mathcal{Q} \to S^2\mathcal{S}^\vee \to 0. $$ Taking into account an exact sequence $$ 0 \to \mathcal{S}^\perp/\mathcal{S} \to \mathcal{Q} \to \mathcal{S}^\vee \to 0 $$ one can obtain an exact sequence $$ 0 \to \mathcal{S}^\vee \otimes (\mathcal{S}^\perp/\mathcal{S}) \to T_{\mathrm{OG}(k,V)} \to \wedge^2\mathcal{S}^\vee \to 0. $$ A similar description exists for the symplectic isotropic Grassmannian.