On the Uniqueness of Semistable Embedding and Domain of Semistable Attraction for Probability Measures on $p$-adic Groups

Abstract

We show that on a $p$-adic Lie group, any normal semistable measure has a unique semistable embedding. This, in particular, implies the uniqueness of semistable embedding of any (operator-)semistable measure on a finite dimensional $p$-adic vector space. We compare two classes of probability measures on a unipotent $p$-adic algebraic group: the class of semistable measures and that of measures whose domain of semistable attraction is nonempty.