# Description of unitary representations of the group of infinite $p$-adic integer matrices

## TL;DR

This paper classifies all irreducible unitary representations of the group of infinite p-adic integer matrices, revealing their structure via finite-dimensional representations of related groups over residue rings.

## Contribution

It provides a complete classification of irreducible unitary representations for infinite p-adic integer matrix groups, a new result in representation theory of infinite p-adic groups.

## Key findings

- Irreducible representations factor through groups over residue rings.
- Representations are induced from finite-dimensional representations of open subgroups.
- The classification applies specifically for p ≠ 2.

## Abstract

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices over a residue ring modulo $p^k$. Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.