Suppose $V\_1$ and $V\_2$ are two $(g,K)$ modules of some reductive group $G$ with maximal compact $K$. Let $P$ be the minimal parabolic of $G$, $U$ its unipotent part, and $u$ its Lie algebra. Suppose the quotients $V\_1/uV\_1$ and $V\_2/uV\_2$ are isomorphic as modules for the Levi component of $P$, then what else do we need to know to conclude that $V\_1$ and $V\_2$ are isomorphic as $G$ modules?


edit:Thanks for Kevin and Emerton's comments and sorry for the confusion about the base field. Here I'm assuming REAL reductive group. 



what information of a representation was killed by Jacquet functor?
This is a comment, mostly on terminology that I think caused some confusion, not an answer, but I don't have enough "influence" to post this as a comment.


For real reductive groups, the Jacquet functor $V\mapsto J(V)$ (Jacquet-Casselman functor, Jacquet module, etc) is defined *differently* from the nonarchimedean case: it is $\varinjlim (V/n^iV)^\*$, which is dual to the n-adic completion of V. Thus J(V) is a p-finite g-module (n is the nilradical, p is the parabolic Lie subalgebra) and the target category of the functor J is parabolic category O for g. On the one hand, this provides more structure: we get a g-module and, for example, the infinitesimal character of V can be read off the infinitesimal character of J(V) (Casselman-Osborne); more generally, V and J(V) have the same annihilator in U(g). On the other hand, J(V) is morally a highest weight module, not a Harish-Chandra module, so some information is lost. The 0th n-homology that this problem is asking about, V/nV, is just the the top layer of J(V), so even more information is lost. By the way, unlike the p-adic Jacquet functor, n-homology is not exact (there may be higher homology), but V/nV is always non-zero (short proof was given by Beilinson and Bernstein).


I assume that the modules V1 and V2 in the formulation were presumed to be simple? 


