for the Verma module $M(\lambda)$, it has a dual $M(\lambda)^{\vee}$,
also as $n^{-}$ module, $M(\lambda)$ isomorphic to $U(n^{-})$
so it is very nature to ask for the dual Verma module $M(\lambda)^{\vee}$ whether 
there is such similar property, like as $n^{+}$ module ,whether there exists an iso??
and how to understand it???



about dual Verma module in BGG category O
As an $\mathfrak n ^+$ modules, all dual Vermas are isomorphic to $U(\mathfrak n^-)$ where we use the coadjoint action, identifying $\mathfrak n^-$ with the dual of $\mathfrak n ^+$ using the killing form.


