I am currently researching the infinite-dimensional unitary representations of the affine group on the field of $p$-adic numbers, defined via the semi direct product
$\mathbb{A}\text{ff}(\mathbb{Q}_p):= \mathbb{Q}_p \rtimes GL(n, \mathbb{Q}_p)$,
meaning this group consists of elements $(a,b)$ for $a,b\in\mathbb{Q}_p$, with group operation, $(a,b)(a',b')=(aa',b'+a'b)$.
I am not sure what the natural topology is on this space - I am not sure how to 'carry' the usual topology on $\mathbb{Q}_p$ through to the semi direct product.
Any help is greatly appreciated
If you are considering $\mathbb{A}\rm{ff}(\mathbb Q_p)$ or $\mathbb{A}\rm{ff} (n,\mathbb Q_p)$ - one has the respective embeddings into matrices, as explained in Wikipedia.
Specifically, for $\mathbb{A}\rm{ff} (n,\mathbb Q_p)$, your elements are pairs $(A,a)$, $A\in\rm{GL}(n,\mathbb Q_p)$, $a\in\mathbb Q_p^n$ and the product is $(A,a)(B,b)=(AB,a+Ab).$
This can be represented as the $(n + 1) × (n + 1)$ block matrix:
${\displaystyle \left({\begin{array}{c|c}A&a\\\hline 0&1\end{array}}\right)}.$
Formally, $\mathbb{A}\rm{ff} (n,\mathbb Q_p)$ is naturally isomorphic to a subgroup of $\rm{GL} (n+1,\mathbb Q_p)$, with $\mathbb Q_p^n$ embedded as the affine plane $\{(v, 1) | v ∈ \mathbb Q_p^n\}$, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the $n × n$ and $1 × 1$ blocks corresponding to the direct sum decomposition $\mathbb Q_p^n⊕\mathbb Q_p$.
The topology on $\rm{GL} (n+1,\mathbb Q_p)$ induce a natural topology on the $\mathbb{A}\rm{ff} (n,\mathbb Q_p)$ (the subspace topology).