# Representations of the Infinite-Dimensional $p$-Adic Affine Group

## TL;DR

This paper introduces an infinite-dimensional p-adic affine group, constructs its irreducible unitary representation, and adapts existing methods to this new context despite the group's non-action on the phase space.

## Contribution

It develops a novel representation theory for an infinite-dimensional p-adic affine group, extending methods used for diffeomorphism groups with necessary modifications.

## Key findings

- Constructed irreducible unitary representations of the p-adic affine group
- Extended representation theory techniques to infinite-dimensional p-adic groups
- Demonstrated the group's action on certain classes of functions

## Abstract

We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.