A structure and representations of diffeomorphism groups of non-Archimedean manifolds

Abstract

Diffeomorphism groups $G$ of manifolds $M$ on locally $\mathbf{F}$-convex spaces over non-Archimedean fields $\mathbf{F}$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex manifolds. It is proved that $G$ is not a Banach-Lie group, but it has a neighbourhood $W$ of the unit element $e$ such that each element $g$ in $W$ belongs to at least one corresponding one-parameter subgroup. It is proved that $G$ is simple and perfect. Its compact subgroups $G_c$ are studied such that a dimension over $\mathbf{F}$ of its tangent space $di{m}_{\mathbf{F}}{T}_e{G}_c$ in $e$ may be infinite. This is used for decompositions of continuous representations into irreducible and investigations of induced representations.

32 pages, Latex