Infinite p-adic random matrices and ergodic decomposition of p-adic Hua measures.

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite p-adic matrices \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right). Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right) that are invariant under the natural action of \mathrm {GL}(\infty,\mathbb {Z}_p)\times \mathrm {GL}(\infty,\mathbb {Z}_p). In this paper we solve the problem of ergodic decomposition for the p-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains. [ABSTRACT FROM AUTHOR]