Let $P$ be a complex parabolic subgroup of a semi-simple complex group $G$. Then we have a decomposition $\mathfrak{g}=\mathfrak{g}_{-k}\oplus \ldots \oplus \mathfrak{g}_0 \oplus \ldots \oplus \mathfrak{g}_k$ respectively to the relative heights , and $[\mathfrak{g}_i,\mathfrak{g}_j]=\{0\} \text{ or } \mathfrak{g}_{i+j}$
The adjoint representation $Ad:P \longrightarrow GL(\mathfrak{g})$ induces a quotient representation $\underline{Ad}:P\longrightarrow GL(\mathfrak{g}/\mathfrak{p})$.
Is the kernel $P'=ker(\underline{ad})$ of this homomorphism known in general ?(In the only examples i can compute ($G=SL_n(\mathbb{C})$) i think this is $P^k = exp(\mathfrak{g}_k)$. But i guess this is not true in general).
Is the following fact true ? Let $G_0=exp(\mathfrak{g}_0)$ the Levi subgroup. Then $G_0$ contains a Cartan subgroup $H=exp_G(\mathfrak{h})$. Moreover, it stabilizes the graduation $(\mathfrak{g}_i)_{i=-k,\ldots,k}$. So if $B\in \mathfrak{p}'=Lie(P')$, then : $B \in \mathfrak{g}^1$, and $ad(g_0)(B) \in \mathfrak{p'}$ for all $g_0 \in G_0$ because $[ad(g_0)(B),ad(g_0)(\mathfrak{g}_-)] = ad(g_0)[B,\mathfrak{g}_-]\subset \mathfrak{p}$.
So i would say that $\mathfrak{p}'= Lie(P')$ is a $H$-submodule, so it's a direct sum of (positive) root spaces $\mathfrak{g}_{\alpha_1}\oplus \ldots \oplus \mathfrak{g}_{\alpha_q}$?
- If 2. is true, can't we construct a holomorphic section $s:P/P' \longrightarrow P$ of the projection $\underline{Ad}$ by considering the complementary roots $\beta_1,\ldots,\beta_p$ in the positive roots, and the application : $$s(\underline{Ad}(exp_G(B_{\beta_1}\oplus \ldots \oplus B_{\beta_p} \oplus B')) = exp_G(B_{\beta_1}\oplus \ldots \oplus B_{\beta_p})$$ where $B_{\beta_i}\in \mathfrak{g}_{\beta_i}$ and $B'\in \mathfrak{p}'$ ?
Best regards