Exercise 12, § 8

a) Let $\lambda , \mu\in \mathfrak{h}^*$. Every non-zero $\mathfrak{g}$-homomorphism from $Z(\mu)$ to $Z(\lambda )$ is injective. (Use Prop. 6 of no. 6.) b) Let $r\in \mathbf{N}, A$ a finite subset of $\mathbf{N}^r,m=$ Card(A). For all $\xi \in \mathbf{N}^r$, let $s(\xi )$ be the sum of the coordinates of $\xi$, and $\mathfrak{P}_A(\xi )$ the number of families $(n_{\alpha})_{\alpha\in A}$ of integers $\geq 0$ such that $\xi =\sum_{\alpha\in A}n_{\alpha}\alpha$. Then $\mathfrak{P}_A(\xi )\leq (s(\xi ) + 1)^m$ for all $\xi \in \mathbf{N}^r$. (Argue by induction on $m.$)

c) Let $\lambda , \mu\in \mathfrak{h}^*$. Show that dim Hom$_{\mathfrak{g}}(Z(\mu),Z(\lambda ))\leq 1$. (Let $\varphi_1$ and $\varphi_2$ be non-zero $\mathfrak{g}$-homomorphisms from $Z(\mu)$ to $Z(\lambda )$. If Im($\varphi_1$) $=$ Im($\varphi_2$)$,\varphi_1$ and $\varphi_2$ are linearly dependent by Prop. 1 (iii) of §6. Assume that Im($\varphi_1$)$\not=$ Im($\varphi_2$). If $Z(\mu)$ is simple, the sum Im($\varphi_1$) $+$ Im($\varphi_2$) is direct; deduce that $\mathfrak{P}(\xi +\lambda -\mu)\geq 2\mathfrak{P}(\xi )$ for all $\xi \in \mathfrak{h}^*$, contradicting $b$). In the general case, use Exerc. 11.)

When dim Hom$_{\mathfrak{g}}(Z(\mu),Z(\lambda )) = 1$, write $Z(\mu)\subset Z(\lambda )$ by abuse of notation. d) Let $\nu \in \mathfrak{h}^*$. The set of $\lambda \in \mathfrak{h}^*$ such that $Z(\lambda -\nu )\subset Z(\lambda )$ is closed in $\mathfrak{h}^*$ in the Zariski topology.