Invariant Sheaves

§0. Introduction The sheaves of tangent vector fields, differential forms or differential operators are canonical. Namely they are invariant by the coordinate transformations. We call such sheaves invariant sheaves. More precisely for a positive integer n, an invariant sheaf on n-manifold is given by the data: coherent OX -module FX for each smooth variety X of dimension n and an isomorphism β(f) : fFY ∼ → FX for any étale morphism f : X → Y. We assume that β(f) satisfies the chain condition (see §1 for the exact definition). The purpose of this paper is to study the properties of invariant sheaves on n-manifold. The first result is that the category I(n) of invariant sheaves is equivalent to the category of modules over a certain group G (with infinite dimension). Let us recall that the category of equivariant sheaves with respect to a transitive action is equivalent to the category of modules over the isotropy subgroup. In our case, manifold may be regarded as a homogeneous space of “the group” of all transformations, and the category of invariant sheaves is regarded as an equivariant sheaf with respect to this action. Let us take an ndimensional vector space V and let G be the group of (formal) transformations that fix the origin. Hence G is a semi-direct product of GLn and a projective limit of finite-dimensional unipotent groups. This G plays a role of the isotropy subgroup and we have