about dual Verma module in BGG category O

Question

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???

Answer 1

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.

Answer 2

As I'm sometimes tempted to do when Ben offers a concise answer, I'll attempt to provide a little wider context to the narrow question being asked. The construction of a Verma module as an induced module is easy and direct, going back more or less to the enveloping algebra methods of Chevalley and Harish-Chandra for proving without case-by-case study that each semisimple Lie algebra over $\mathbb{C}$ has a simple module of arbitrary (at first dominant integral) highest weight. Here the required simple module occurs in principle as the unique simple quotient of a Verma module, a construction which works for arbitrary highest weights. The induction process and PBW theorem make it transparent that a Verma module is isomorphic as $U(\mathfrak{n}^-)$-module to $U(\mathfrak{n}^-)$. Moreover, the formal character of this module relative to the fixed Cartan subalgebra is obvious.

But nothing is revealed about the multiplicities or formal characters of the finitely many composition factors of a Verma module. Verma got started on the more systematic study of that hard problem, but only the later work of BGG and then Kazhdan-Lusztig led to a complete solution. Besides using Verma modules in a nice way to recover the classical character formulas of Weyl and Kostant for finite dimensional modules, BGG opened the way to the use of categorical methods and homological algebra. But in their basic category one can't do straightforward contructions using for example co-induction and vector space duality. Their duality functor is more special, leading many people to use the symbolism $M^\vee$ (rather than say $M^*$). Here it's easy enough to characterize the $U(\mathfrak{n}^-)$-module underlying $M(\lambda)^\vee$, which has the same formal character but an "upside down" module structure. Here the "raising" and "lowering" operators essentially get switched. But no new module information emerges.

For BGG and others the main use of duality has been in homological manipulations involving tensor products and the Hom functor. So even though the question asked is meaningful (if phrased a bit more carefully), it doesn't seem to lead to better insights.