Ergodic measures on infinite skew-symmetric matrices over non-Archimedean local fields

Abstract

Let $F$ be a non-discrete non-Archimedean locally compact field such that the characteristic $\mathrm{ch}(F)\ne 2$ and let ${\mathcal{O}}_F$ be the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Skew}(\mathbb{N},F)$ of infinite skew-symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty ,{\mathcal{O}}_F)$ and Theorem 1.4, that gives an unexpected natural correspondence between the set of $\mathrm{GL}(\infty ,{\mathcal{O}}_F)$-invariant Borel probability measures on $\mathrm{Sym}(\mathbb{N},F)$ with the set of $\mathrm{GL}(\infty ,{\mathcal{O}}_F)\times \mathrm{GL}(\infty ,{\mathcal{O}}_F)$-invariant Borel probability measures on the space $\mathrm{Mat}(\mathbb{N},F)$ of infinite matrices over $F$.

Skew-symmetric case of similar results in a previous paper, with simplified presentation. 12 pages