can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly?

Question

Can we write down the holomorphic vector fields on the compact hermitian symmetric spaces explicitly? Do you have any idea of which paper has disscussed this topic? For example, what is the dimension of the vector space of holomorphic vector fields on a grassmanian?

Answer 1

I suggest that you look at the treatment in Helgason's classic book "Differential Geometry, Lie Groups, and Symmetric Spaces".

The result you seek is that the holomorphic vector fields on a compact Hermitian symmetric space $M = U/K$ are the complex Lie algebra spanned by the vector fields of the form $Z = X - i JX$ where $X$ is a Killing vector field of the $U$-invariant metric on $M$. In particular, $U$ is a maximal compact subgroup of a holomorphic Lie group $G$ and $M=G/P$ where $P$ is a parabolic subgroup of $G$. The holomorphic transformations of $M$ are simply those generated by the left action of $G$.

For example, $\mathbb{CP}^n = \mathrm{SU}(n{+}1)/\mathrm{U}(n) = \mathrm{SL}(n{+}1,\mathbb{C})/P$ where $P$ is the subgroup of $\mathrm{SL}(n{+}1,\mathbb{C})$ that fixes a line $\mathbb{C}\subset\mathbb{C}^{n+1}$.

To answer your specific question, the space of holomorphic vector fields on the Grassmannian $\mathrm{Gr}(k,\mathbb{C}^n) = \mathrm{SL}(n,\mathbb{C})/P_k$ (where $P_k$ is the subgroup of special linear transformations of $\mathbb{C}^n$ that preserves a fixed subspace $\mathbb{C}^k\subset\mathbb{C}^n$) is isomorphic, as a Lie algebra, to $\mathfrak{sl}(n,\mathbb{C})$ and hence has (complex) dimension $n^2{-}1$.