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 $G$.
Theorem. [Kashiwara-Schmid (?)]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.
[More generally, can one use###[Added later]###
Emerton (below) mentions the BeilinsonSchmid-Bernstein correspondenceVilonen picture, which appears to be very well adapted to the study of our poset $\mathbf{Glob}(G)$. Let me at introduce the structuremain ideas of the posetobjects of globalizationsinterest, 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 languageflag manifold of the complexification of $K$$G$. Let $\lambda\in\mathfrak{h}^{\vee}$ be a dominant element of the dual space of the universal Cartan; for simplicity, let's assume that it is regular. Now one can define the twisted equivariant bounded derived categories $D^b_G(X)_{-\lambda}$ and $D^b_K(X)_{-\lambda}$ of constructible sheaves on $X$. Now let $\mathbf{Glob}(G,\lambda)$ denote the poset of globalizations for admissible representations with infinitesimal character $\chi_{\lambda}$, so the objects are exact functors $\mathcal{G}\colon\mathbf{HC}(\mathfrak{g},K)_{\chi_{\lambda}}\to\mathbf{Rep}(G)_{\chi_{\lambda}}$ equipped with natural isomorphisms $\eta_{\mathcal{G}}:\mathcal{M}\circ\mathcal{G}\simeq\mathrm{id}$.
Matsuki correspondence. [Mirkovic-equivariantUzawa-Vilonen] There is a canonical equivalence $\mathcal{D}$$\Phi\colon D^b_G(X)_{-\lambda}\simeq D^b_K(X)_{-\lambda}$. The perverse t-modulesstructure 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 flag manifold? Ifhearts.
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 restrictnow expect that one can show the following $\mathcal{M}$(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, sayevery 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, Fréchet nuclear spacesnote that if my expectation holds, how much smaller does thisthen one should be able to find a copy of the poset become?]$\mathbf{Glob}(G,\lambda)$ embedded in $D^b_G(X)_{-\lambda}$.
