# About Hom and weight space of nilpotent Lie algebra cohomology

Let $$\mathfrak{g}$$ be a complex semisimple Lie algebra. Denote by $$\Phi$$ the root system of $$(\mathfrak{g},\mathfrak{h})$$ and denote by $$\mathfrak{g}_\alpha$$ the root subspace of $$\mathfrak{g}$$ corresponding to a root $$\alpha$$.

We fix a choice of the set of positive roots $$\Phi^+$$, and let $$\Delta$$ be the corresponding subset of simple roots in $$\Phi^+$$. Note that each subset $$I\subseteq\Delta$$ generates a root system $$\Phi_I\subseteq\Phi$$, with positive roots $$\Phi_I^+=\Phi_I\cap \Phi^+$$.

There are a number of subalgebras of $$\mathfrak{g}$$ associated with the root system $$\Phi_I$$. Let $$\mathfrak{l}_I=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_I}\mathfrak{g}_\alpha$$ be the Levi subalgebra and let $$\mathfrak{u}_I=\sum_{\alpha\in\Phi^+\backslash\Phi_I^+}\mathfrak{g}_\alpha$$ be the nilpotent radical.

We note that once $$I$$ is fixed, there is little use for other subsets of $$\Delta$$. Therefore, we omit the subscript if a subalgebra is obviously associated to $$I$$.

Denote $$N_\mu$$ the $$\mu$$-weight space of $$\mathfrak{l}$$-module $$N$$. Let $$F(\lambda)$$ is the finite dimensional simple $$\mathfrak{l}$$-module with highest weight $$\lambda$$.

1. Does $$\text{Hom}_\mathfrak{l}(F(\lambda), H^i(\mathfrak{u},V))\cong H^i(\mathfrak{u},V)_{\lambda}$$?

This is true for general modules, you don't need anything specific about nilpotent cohomology. Every homomorphism from a $$F(\lambda)$$ is uniquely determined by it's highest weight vector.