Let $ mathfrak {g} $ to be a complex semisimple 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 onedimensional $ mathfrak {b} $module.

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

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

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