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?