Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the proalgebraic 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?
1 Answer
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Assume X is ndimensional and regular. Then there is a functor from Gmodules V to D_{X}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 G_{0} and P with the canonical G_{0}torsor, the same construction yields an O_{X}module. The extra structure of a Gaction lets you identify infinitesimally nearby fibers.

$\begingroup$ I'm confused. Shouldn't you need the the action of a HarishChandra pair (roughly since k[[x_1,...,x_n]] has derivations that don't integrate)? Group actions shouldn't be able to glue infinitesimally close points. Or is this some subtle thing where thinking about the algebraic group encapsulates the Lie algebra bit? $\endgroup$– Ben Webster ♦Commented Oct 14, 2009 at 22:49

1$\begingroup$ You can formally exponentiate a HarishChandra pair. G_0 is a proalgebraic group, but G is a formal group with a big proalgebraic subgroup. You can think of it as a group indscheme. $\endgroup$– S. Carnahan ♦Commented Oct 14, 2009 at 23:01

$\begingroup$ So.... that would be a yes. $\endgroup$– Ben Webster ♦Commented Oct 15, 2009 at 12:55

$\begingroup$ It's sort of a yes for both questions, but you change "algebraic" to "formal" in the second. You can think of G as G_0 with extra fuzz. $\endgroup$– S. Carnahan ♦Commented Oct 15, 2009 at 15:31