Infinitesimal equivalence of admissible representations

Question

Let $G_0$ be the $\mathbb{R}$-points of a real reductive group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. What is the precise relation between the category of admissible representations of $G_0$ and the category of admissible $(\mathfrak{g},K)$-modules? If we restrict to unitary (admissible) representations and $(\mathfrak{g},K)$-modules, they are supposed to be equivalent, but what about non-unitary representations? Are there counterexamples for e.g. $\text{SL}_2(\mathbb{R})$?

Answer 1

Probably the relevant work is that of Casselman and Wallach (independently) on (in effect) adjoints (right? left?) to the forgetful functor taking Lie group repns to Lie algebra repns. The keyword is "globalization". It turns out that a right adjoint is not the left adjoint (which could be anticipated by observing that many different Lie group repns have the same "smooth vectors", already for the Lie group $SO(2,\mathbb R)$.

So, within mild constraints, it really is true that there exists (more than one) "globalization" of a $(\mathfrak g,K)$ repn, because the worst obstruction is just the topology of $K$.

The papers of Casselman, and of Wallach, should be easily visible on MathSciNet. If it seems otherwise, I can add information here.