# A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fields

## TL;DR

This paper constructs a canonical way to decompose irreducible representations of the multiplicative group of a division algebra over a local field into one-dimensional projective bases, advancing understanding of their structure.

## Contribution

It introduces a method to explicitly construct projective bases for irreducible representations of these groups, a novel approach in the representation theory of division algebras over local fields.

## Key findings

- Canonical decomposition of irreducible representations into one-dimensional subspaces
- Explicit construction of projective bases for these representations
- Enhanced understanding of the structure of multiplicative groups of division algebras

## Abstract

Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $ G=D^{\star}$ be the multiplicative group of $D$. The goal of this paper is to provide a canonical decomposition of any complex irreducible representation $V$ of $G$ in a direct sum of one-dimensional subspaces.