Question about $\text{Hom}_{\mathfrak{g}}(M,L)$

Let $$\mathfrak{g}$$ be a complex semisimple Lie algebra.

Suppose $$M,N,L$$ are $$\mathfrak{g}$$-modules, $$N$$ is a $$\mathfrak{g}$$-submodule of $$M$$.

Does this implies $$\text{Hom}_{\mathfrak{g}}(N,L)\le \text{Hom}_{\mathfrak{g}}(M,L)$$ as a vector subspace?

If yes, how to prove it?

Firstly, since the functor Hom is contravariant in the first variable, what you expect is a map going the other way. For a general abelian category, the only situation in which you expect to have an inclusion $$\mathrm{Hom}(A,B) \subseteq \mathrm{Hom}(C,B)$$ is when $$A$$ is a quotient of $$C$$. For a semi-simple Lie algebra $$\mathfrak{g}$$ over $$\mathbf{C}$$, the category of finite-dimensional modules is semi-simple, so there are (in general, many) ways to represent a given submodule $$N \subseteq M$$ as a quotient. Choosing one of them gives you the desired inclusion (which is not canonical in general!).
However, the category of all $$\mathfrak{g}$$-modules is far from semi-simple, so there is no such inclusion in the generality you are asking for. For instance, if $$\mathfrak{g}=\mathfrak{sl}_2(\mathbf{C})$$ and you take $$N=L=\mathbf{C}$$ to be the trivial representation and $$M$$ to be the co-Verma module containing it as a submodule, then $$\mathrm{Hom}_\mathfrak{g}(M,\mathbf{C})=0,$$ so there can be no inclusion as in your question.
• But how to represent a given submodule $N\subseteq M$ as a quotient of $M$ in the semisimple Lie algebra case? Commented Nov 27, 2018 at 14:21
• @JamesCheung Are your representations finite dimensional $\mathbf{C}$-vector spaces? If so, you must choose a complementary submodule. There are many ways to do so, in general. It's impossible for me to specify one of them canonically, as I noted in my answer. One procedure would be to start with a positive definite Hermitian form on $M$ and to average it over the special unitary group to get an invariant positive definite form, which you could then use by taking orthogonal complements. Commented Nov 27, 2018 at 14:23