Exercise 16, § 7

Let V and $V'$ be two $\mathfrak{g}$-modules. Then $V'$ is said to be subordinate to V if there exists a linear map $f: V\rightarrow V'$ such that:

$\alpha )f$ is surjective; $\beta )$ the image under $f$ of a primitive element of V is either 0 or a primitive element of $V';\gamma )f$ is an $\mathfrak{n}_-$-homomorphism.

a) Let $f: V\rightarrow V'$ satisfy $\alpha ),\beta ),\gamma )$. Let $v$ be a primitive element of V. Then the image under $f$ of the $\mathfrak{g}$-submodule generated by $v$ is the $\mathfrak{g}$-submodule generated by $f(v)$.

b) Let $f: V\rightarrow V'$ satisfy $\alpha ),\beta ),\gamma )$. Let W be a $\mathfrak{g}$-submodule of V. Then $f(W)$ is a $\mathfrak{g}$-submodule of $V'$ that is subordinate to W.

c) Let $V = E_1\oplus \cdots \oplus E_s$ and $V'= E'_1\oplus \cdots E'_{s'}$ be decompositions of V and $V'$ into sums of simple modules. Then $V'$ is subordinate to V if and only if $s'\leq s$ and there exists $\sigma \in \mathfrak{S}_s$ such that $E'_i$ is subordinate to $E_{\sigma(i)}$ for $i= 1, . . . , s'$.

d) If $V'$ is subordinate to V and if V is simple, then $V'$ is simple or reduced to 0.

e) Assume that V and $V'$ are simple. Let $\lambda$ and $\lambda '$ be their highest weights. Then $V'$ is subordinate to V if and only if $\lambda '(H_{\alpha})\leq \lambda (H_{\alpha})$ for all $\alpha \in B$. (For the sufficiency, use the preceding exercise.)