Spaces of smooth functions and distributions on infinite-dimensional compact groups

Several scales of smooth functions are introduced in the setting of connected infinite-dimensional compact groups. These are spaces of functions on the group with continuous derivatives in certain directions. We study properties of these spaces and of associated distribution spaces. Some of these spaces are intrinsically associated with the infinitesimal generator of a given Gaussian convolution semigroup. One of the reasons for studying these smooth function and distribution spaces is to obtain sharp results concerning the hypoellipticity of the infinitesimal generators of Gaussian convolution semigroups, i.e., invariant sub-Laplacians on compact groups.