Measures on diffeomorphism groups for non-Archimedean manifolds: Group representations and their applications

Nondegenerate $\sigma$-additive measures with ranges in $\mathbb R$ and $\mathbb Q_q$ ($q\ne p$ are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups $G'$ are constructed on diffeomorphism and homeomorphism groups $G$ for separable non-Archimedean Banach manifolds $M$ over a local field $\mathbb K$, $\mathbb K\supset\mathbb Q_p$, where $\mathbb Q_p$ is the field of $p$-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory.