Provarieties and observable subgroups of pro-affine algebraic groups

Abstract

We define a closed subgroup H of a pro–affine algebraic group G to be observable if every (finite dimensional) rational H module is an H submodule of a rational G module. We show the equivalence to observability of a number of other conditions on the pair G,H, including the condition that the provariety G/H be pro–quasi–affine. To treat this latter, we also include results on the foundations of provarieties.