# Tangent bundle for orthogonal and isotropic Grassmannians

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.

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.
• The second sequence is a combination of the tautological sequence $$0 \to \mathcal{S} \to V \otimes \mathcal{O} \to \mathcal{Q} \to 0$$ and of its dual $$0 \to \mathcal{S}^\perp \to V \otimes \mathcal{O} \to \mathcal{S}^\vee \to 0$$. Commented Aug 19, 2021 at 9:19