  Let $p$ be a prime number, $V$ a discrete valuation ring of unequal
caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$.
Using partial divided powers techniques of Berthelot, we construct arithmetic
distribution algebras, with level $m$, generalizing the classical construction
of the distribution algebra. We also construct the weak completion of the
classical distribution algebra. We then show that these distribution algebras
can be identified with invariant arithmetic differential operators over $G$. We
finally apply these constructions in the case of a reductive group and obtain a
localization theorem for the sheaf of arithmetic differential operators on the
formal flag variety obtained by $p$-adic completion, generalizing a previous
result of Ardakov-Wadsley (for the level 0 and with some condition on $p$).
