Formal Geometry

I am writing to post an answer to my own question. The answer below consists of what I was able to jot down from a seminar talk by Jacob Lurie, and a patient followup explanation by Roman Travkin, followed by a correction by David Ben-Zvi. Of course, mistakes and naivete in the translation are solely attributed to me. Since the answer was given to me in response to my asking on MO, it seems that karma dictates that I record it here.

The deRham stack, $X_{dR}$, of a scheme $X$ is defined as a functor on test schemes S by

$$X_{dR}(S) := {textrm{Maps}: S_{red} to X }.$$

where $S_{red}$ is the reduced scheme associated to a scheme $S$. It is representable as a stack - I think always, but at least when $X$ is smooth - which we should assume for later purposes anyways (maybe everything should be based over $mathbb{C}$ also? I welcome corrections, which I'll incorporate).

Okay so $X$, viewed as the functor $X(S)={textrm{Maps}:Sto X}$ has a natural map $pi:Xto X_{dR}$, by pre-composing with $S_{red} to S$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(xin X, t_x: X_{(x)}cong hat{D_n})$, where $hat{D_n}$ is the formal disc, $mathcal{O}(hat{D_n}):=mathbb{C}[[x_1,ldots,x_n]]$. The fiber over each point $xin X$ is the group $Aut^0(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(hat{D_n})$, the group of automorphisms of $hat{D_n}$ preserving the origin.

Now, the scheme $Aut_X$ is actually a $hat G$-torsor over $X_{dR}$, where (I gather from David Ben-Zvi's comments) $hat G$ is something more like an inductive limit of the groups of automorphisms on $n$th infinitessimal neighboroods of the origin in $D_n$.

Now, suppose that $M$ is a $W_n$ module, which has the property that the operators $x_ipartial_i$ for all $i$ are diagonalizable with finite dimensional eigenspaces. (These we should think of as being the diagonals $h_i$ for a copy of $mathfrak{gl}_n$ sitting inside $W_n$). In this case, one can prove some nice things.

First, in representation theory: $M$ lies in the category generated by modules $mathcal{F}_lambda$ which are coinduced from $V_lambda$, the irreducible of $mathfrak{gl}_n$.` That is, as vector spaces the $mathcal{F}_lambda$s are just $V_lambda otimes C[x_1,ldots, x_n]$, and the action is given by natural formulas. So $M$ is a (possibly infinite) extension of such things. But there is a theory of weights which control the representation theory somewhat. See A. N. Rudakov. Irreducible representations of infinite-dimensional Lie algebras of Cartan type. Math. USSR Izv. Vol. 8, pgs. 836-866, which has been translated to English.

Second, in geometry: The module $M$ is a Harish-Chandra module for the pair $(W_n,Aut_0(C[[x_1,...,x_n]]))$. This means: $W_n$ has a Lie sub algebra $W^0$ of vector fields which vanish at the origin (so they have no constant vector field terms like $partial_i$). $W^0$ (or perhaps its completion w.r.t order of vanishing at origin) is the Lie algebra of $Aut_0(C[[x_1,...,x_n]])$, as it consists of derivations which preserve the maximal ideal (not sure exactly how you make precise that it is "the Lie algebra", but anyways, there is an exponential map turning an integrable $W^0$ module into an $Aut_0(C[[x_1,...,x_n]]$)-module ). The assumptions on $M$ were precisely those that make $M$ an integrable $W^0$ module, and so we can regard $M$ as a $G$-module. Well now we have an associated bundle construction for the $G$-torsor $Aut_Xto X$, and we can use this to produce a sheaf of vector spaces (not quasi-coherent!) over $X$ with fiber $M$.

Better $M$ has an action of the operators $partial_i$, and we can exponentiate the action of all of $W_n$ to the group $hat G$. That means that we can instead construct the associated bundle for the $hat G$ torsor $Aut_Xto X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's) definition is a crystal of vector spaces on $X$.