Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie group. Denote by $\mathfrak{g}$ the complexified Lie algebra, and denote by $K$ a maximal compact in the complexification of $G$.
Denote by $\mathbf{HC}(\mathfrak{g},K)$ the category of admissible $(\mathfrak{g},K)$-modules or (Harish Chandra modules), and $(\mathfrak{g},K)$-module homomorphisms. Denote by $\mathbf{Rep}(G)$ the category of admissible representations of finite length (on complete locally convex Hausdorff topological vector spaces), with continuous linear $G$-maps.
The Harish Chandra functor $\mathcal{M}\colon\mathbf{Rep}(G)\to\mathbf{HC}(\mathfrak{g},K)$ assigns to any admissible representation $V$ the Harish Chandra module of $K$-finite vectors of $V$. This is a faithful, exact functor. Let us call an exact functor $\mathcal{G}\colon\mathbf{HC}(\mathfrak{g},K)\to\mathbf{Rep}(G)$ along with a comparison isomorphism $\eta_{\mathcal{G}}\colon\mathcal{M}\circ\mathcal{G}\simeq\mathrm{id}$ a globalization functor.
Our first observation is that globalization functors exist.
Theorem. [Casselman-Wallach] The restriction of $\mathcal{M}$ to the full subcategory of smooth admissible Fréchet spaces is an equivalence. Moreover, for any Harish Chandra module $M$, the essentially unique smooth admissible representation $(\pi,V)$ such that $M\cong\mathcal{M}(\pi,V)$ has the property $\pi(\mathcal{S}(G))M=V$, where $\mathcal{S}(G)$ is the Schwartz algebra of $G$.
If we do not restrict $\mathcal{M}$ to smooth admissible Fréchet spaces, then we have a minimal globalization and a maximal one.
Theorem. [Kashiwara-Schmid] $\mathcal{M}$ admits both a left adjoint $\mathcal{G}_0$ and right adjoint $\mathcal{G}_{\infty}$, and the counit and unit give these functors the structure of globalization functors.
Construction. Here, briefly, are descriptions of the minimal and maximal globalizations. The minimal globalization is
$$\mathcal{G}_0=\textit{Dist}_c(G)\otimes_{U(\mathfrak{g})}-$$
where $\textit{Dist}_c(G)$ denotes the space of compactly supported distributions on $G$, and the maximal one is
$$\mathcal{G}_{\infty}=\mathrm{Hom}_{U(\mathfrak{g})}((-)^{\vee},C^{\infty}(G))$$
where $M^{\vee}$ is the dual Harish Chandra module of $M$ (i.e., the $K$-finite vectors of the algebraic dual of $M$).
For any Harish Chandra module $M$, the minimal globalization $\mathcal{G}_0(M)$ is a dual Fréchet nuclear space, and the maximal globalization $\mathcal{G}_{\infty}(M)$ is a Fréchet nuclear space.
Example. If $P\subset G$ is a parabolic subgroup, then the space $L^2(G/P)$ of $L^2$-functions on the homogeneous space $G/P$ is an admissible representation, and $M=\mathcal{M}(L^2(G/P))$ is a particularly interesting Harish Chandra module. In this case, one may identify $\mathcal{G}_0(M)$ with the real analytic functions on $G/P$, and one may identify $\mathcal{G}_{\infty}(M)$ with the hyperfunctions on $G/P$.
[I think other globalizations with different properties are known or expected; I don't yet know much about these, however.]
Consider the category $\mathbf{Glob}(G)$ of globalization functors for $G$; morphisms $\mathcal{G}'\to\mathcal{G}$ are natural transformations that are required to be compatible with the comparison isomorphisms $\eta_{\mathcal{G}'}$ and $\eta_{\mathcal{G}}$. Since $\mathcal{M}$ is faithful, this category is actually a poset, and it has both an inf and a sup, namely $\mathcal{G}_0$ and $\mathcal{G}_{\infty}$. This is the poset of globalizations for $G$.
I'd like to know more about the structure of the poset $\mathbf{Glob}(G)$ — really, anything at all, but let me ask the following concrete question.
Question. Does every finite collection of elements of $\mathbf{Glob}(G)$ admit both an inf and a sup?
[Added later]
Emerton (below) mentions a geometric picture that appears to be very well adapted to the study of our poset $\mathbf{Glob}(G)$. Let me at introduce the main ideas of the objects of interest, and what I learned about our poset. [What I'm going to say was essentially outlined by Kashiwara in 1987.] For this, we probably need to assume that $G$ is connected.
Notation. Let $X$ be the flag manifold of the complexification of $G$. Let $\lambda\in\mathfrak{h}^{\vee}$ be a dominant element of the dual space of the universal Cartan; for simplicity, let's assume that it is regular. Now one can define the twisted equivariant bounded derived categories $D^b_G(X)_{-\lambda}$ and $D^b_K(X)_{-\lambda}$ of constructible sheaves on $X$. Now let $\mathbf{Glob}(G,\lambda)$ denote the poset of globalizations for admissible representations with infinitesimal character $\chi_{\lambda}$, so the objects are exact functors $\mathcal{G}\colon\mathbf{HC}(\mathfrak{g},K)_{\chi_{\lambda}}\to\mathbf{Rep}(G)_{\chi_{\lambda}}$ equipped with natural isomorphisms $\eta_{\mathcal{G}}:\mathcal{M}\circ\mathcal{G}\simeq\mathrm{id}$.
Matsuki correspondence. [Mirkovic-Uzawa-Vilonen] There is a canonical equivalence $\Phi\colon D^b_G(X)_{-\lambda}\simeq D^b_K(X)_{-\lambda}$. The perverse t-structure on the latter can be lifted along this correspondence to obtain a t-structure on $D^b_G(X)_{-\lambda}$ as well. The Matsuki correspondence then restricts to an equivalence $\Phi\colon P_G(X)_{-\lambda}\simeq P_K(X)_{-\lambda}$ between the corresponding hearts.
Beilinson-Bernstein construction. There is a canonical equivalence $\alpha\colon P_K(X)_{-\lambda}\simeq\mathbf{HC}(\mathfrak{g},K)_{\chi_{\lambda}}$, given by Riemann-Hilbert, followed by taking cohomology. [If $\lambda$ is not regular, then this isn't quite an equivalence.]
Now we deduce a geometric description of an object of $\mathbf{Glob}(G,\lambda)$ as an exact functor $\mathcal{H}\colon P_G(X)_{-\lambda}\to\mathbf{Rep}(G)_{\chi_{\lambda}}$ equipped with a natural isomorphism $\mathcal{M}\circ\mathcal{H}\simeq\alpha\circ\Phi$, or equivalently, as a suitably t-exact functor $\mathcal{H}\colon D^b_G(X)_{-\lambda}\to D^b\mathbf{Rep}(G)_{\chi_{\lambda}}$ equipped with a functorial identification between the (complex of) Harish Chandra module(s) of $K$-finite vectors of $\mathcal{H}(F)$ and $\mathrm{RHom}(\mathbf{D}\Phi F,\mathcal{O}_X(\lambda))$ for any $F\in D^b_G(X)_{-\lambda}$. In particular, as Emerton observes, the maximal and minimal globalizations can be expressed as
$$\mathcal{H}_{\infty}(F)=\mathrm{RHom}(\mathbf{D}F,\mathcal{O}_X(\lambda))$$
and
$$\mathcal{H}_0(F)=F\otimes^L\mathcal{O}_X(\lambda)$$
Note that Verdier duality gives rise to an anti-involution $\mathcal{H}\mapsto(\mathcal{H}\circ\mathbf{D})^{\vee}$ of the poset $\mathbf{Glob}(G,\lambda)$; in particular, it exchanges $\mathcal{H}_{\infty}$ and $\mathcal{H}_0$.
I now expect that one can show the following (though I don't claim to have thought about this point carefully enough to call it a proposition).
Conjecture. All globalization functors are representable. That is, every element of $\mathbf{Glob}(G,\lambda)$ is of the form $\mathrm{RHom}(\mathbf{D}(-),E)$ for some object $E\in D^b_G(X)_{-\lambda}$.
Question. Can one characterize those objects $E\in D^b_G(X)_{-\lambda}$ such that the functor $\mathrm{RHom}(\mathbf{D}(-),E)$ is a globalization functor? Given a map between any two of these, under what circumstances do they induce a morphism of globalization functors (as defined above)?
In particular, note that if my expectation holds, then one should be able to find a copy of the poset $\mathbf{Glob}(G,\lambda)$ embedded in $D^b_G(X)_{-\lambda}$.
