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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?

flag	asked Oct 14 2009 at 22:01 Ian Shipman 595●2●8

S. Carnahan · Accepted Answer · 2009-10-15 04:28:18Z UTC

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Assume X is n-dimensional and regular. Then there is a functor from G-modules 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 G₀ and P with the canonical G₀-torsor, the same construction yields an O_X-module. The extra structure of a G-action lets you identify infinitesimally nearby fibers.

edited Oct 15 2009 at 4:28

answered Oct 14 2009 at 22:34

S. Carnahan♦
25.6k●41●90

	I'm confused. Shouldn't you need the the action of a Harish-Chandra 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? – Ben Webster♦ Oct 14 2009 at 22:49
	You can formally exponentiate a Harish-Chandra 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 ind-scheme. – S. Carnahan♦ Oct 14 2009 at 23:01
	So.... that would be a yes. – Ben Webster♦ Oct 15 2009 at 12:55
	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. – S. Carnahan♦ Oct 15 2009 at 15:31

what is the connection between D-modules and coordinate bundles?

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