# On the group of infinite $p$-adic matrices with integer elements

## TL;DR

This paper studies infinite-dimensional $p$-adic matrix groups with integer entries, demonstrating that their unitary representations can be extended to associated double coset categories, generalizing known results from real classical groups.

## Contribution

It proves that unitary representations of infinite $p$-adic matrix groups automatically extend to their train categories, extending the theory from real to $p$-adic groups.

## Key findings

- Unitary representations extend to the train category for infinite $p$-adic groups.
- A key lemma about the complete group of infinite $p$-adic matrices with integer coefficients.
- Generalization of automatic extension phenomena from real to $p$-adic groups.

## Abstract

Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train) and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma about the complete group $GL$ of infinite $p$-adic matrices with integer coefficients, this lemma implies that the phenomenon of automatic extension of unitary representations to trains is valid for infinite-dimensional $p$-adic groups.