Linear representation theory of general affine group of degree one over a finite field

This article discusses the linear representation theory of the general affine group of degree one over a finite field of size $$q$$ and characteristic $$p$$, where $$q = p^r$$.

Irreducible representations
We denote by the pair $$(a,\mu)$$ the element $$x \mapsto a + \mu x$$. The multiplicative part is $$\mu \in \mathbb{F}_q^\ast$$ and the additive part is $$a \in \mathbb{F}_q$$.