TITLE: what is the connection between D-modules and coordinate bundles?
QUESTION [8 upvotes]: Fix $n$ and a field $k$ of characteristic zero.  Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$.  Let $G_0$ be the subgroup of automorphisms preserving the closed point  (note that for general $T$, $G_0(T)$ can be a proper subgroup of $G(T)$).  Let $X$ be a regular variety over $k$ and let $P$ be the principal $G$ bundle of formal coordinate systems, naturally a $G$ torsor over $X$.  I hear that there is a connection between $P$ and $D_X$-modules.  what is this connection?

REPLY [4 votes]: Assume X is n-dimensional and regular.  Then there is a functor from G-modules V to DX-modules, given by an associated bundle construction.  Take the trivial (ind-)bundle on P with fiber V, and quotient by the action of G on P and V.  If you replace G with G0 and P with the canonical G0-torsor, the same construction yields an OX-module.  The extra structure of a G-action lets you identify infinitesimally nearby fibers.