# About Hom and the weight space of cohomology

Let $$mathfrak {g}$$ to be a complex semi-simple Lie algebra, $$mathfrak {n}$$ to be the maximum nilpotent subalgebra in Borel subalgebra $$mathfrak {b}$$.

Denote $$N_ mu$$ the $$mu$$-space of $$mathfrak {h}$$-module $$N$$.
Let $$mathbb {C} _ { lambda}$$ is one-dimensional $$mathfrak {b}$$-module.

1. Is $$text {Hom} _ mathfrak {h} ( mathbb {C} _ { lambda}, H ^ i ( mathfrak {n}, V)) cong H ^ i ( mathfrak {n}, V ) _ { lambda}$$?

2. Is $$H ^ 0 ( mathfrak {n}, V) _ lambda cong V_ lambda$$ as a vector space? I think that's true, but how to prove it?

3. Is $$left (V ^ mathfrak {n} right) _ lambda cong V_ lambda$$ as a vector space?