Topological transformation groups of manifolds over non-Archimedean fields, their representations, and quasi-invariant measures. I

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Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups.