# How many ways are there to globalize Harish Chandra modules?

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?

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 $D^b_G(X)_{-\lambda}$.

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Yesterday, while doing my background research on globalizations, I stumpled upon not only this question but also on lecture notes by Schmid. If I understand your (first) question correctly, this quote should be helpful: "Banach globalizations, we should remark, cannot be functorial, as can be seen already in simple examples." goo.gl/QVq8Z –  Vít Tuček Feb 20 '12 at 8:54

Dear Clark,

This answer is addressed primarily at the final paranthetical comment. There is a "dual" geometric description of $(\mathfrak g, K)$-modules to the Beilinson--Bernstein picture, using orbits of the real group $G(\mathbb R)$ on the flag variety. It is employed by Schmid and Vilonen in many of their papers, and there is an expository article about it by Vilonen in one of the Park City proceedings.

In this dual picture, when one takes sections of the sheaves, one really gets $G(\mathbb R)$-representations; I'm not sure off the top of my head which models you get though (the smooth ones or some other ones, or whether you get an option depending on the particular brand of sheaves you use).

Regarding your broader question, I've always imagined that one can find a model for a $(\mathfrak g, K)$-module using any brand of regularity you choose (analytic, smooth, distributional, hyperfunctions, and other brands in between, whatever they might be (and perhaps answering that is part of the point of the question!)), and that these will be ordered in the obvious way. But this is certainly not the precise answer you want, and is only a vague intuition.

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Hi Matthew - Thanks for your answer! (It sounds to me as though your "vague intuition" is exactly the sort of thing I've been trying to formalize.) I should have mentioned the Schmid-Vilonen description in the body my question; you seem to be suggesting that their picture might be better adapted to my question. (In particular, I better continue my education on the subject!) Should I be seeking a dictionary between globalization functors and certain fibered categories of sheaves or ind-sheaves on the flag variety? or is this too much to hope for? –  Clark Barwick Feb 11 '10 at 3:50
Dear Clark, I'm not sure about your last question. I am pretty confident though that the Schmid-Vilonen description is more amenable to these sorts of analytic questions. –  Emerton Feb 11 '10 at 4:19
Dear Clark, I spoke with Vilonen this morning, and learnt the following: if we work in the Schmid-Vilonen setting, then taking cohomology gives the minimal (analytic) model, and a dual construction (phrased as an appropriate Ext) gives the maximal (hyperfunction) model. You probably knew this already, though, since I think it is in Kashiwara-Schmid. –  Emerton Feb 11 '10 at 16:42
Matthew - Thanks again for your insight! I thought a bit more about your comments, and I modified my question accordingly. Your observation was very helpful. –  Clark Barwick Feb 13 '10 at 16:53
Dear Clark, You're very welcome --- I'm glad it was of some help. –  Emerton Feb 13 '10 at 17:39