The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Question

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers.

We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under translation by some open subgroup.

Let $C_c^\infty(\mathbb{Q}_p^*)$ be the space of (complex-valued) smooth compactly supported functions on $\mathbb{Q}_p^*$. Equivalently, $C_c^\infty(\mathbb{Q}_p^*)$ is the space of locally constant and compactly supported functions on $\mathbb{Q}_p^*$ (locally constant = continuous relative to the discrete topology on $\mathbb{C}$).

I would like to visualize the elements of $C_c^\infty(\mathbb{Q}_p^*)$, and understand how taking their integral works. I thought about it for a while, and I'm looking for someone who is familiar with this to verify what I did (and possibly give some advice).

I use the direct product decomposition $\mathbb{Q}_p^*=\langle p\rangle\cdot \mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p)$.

So, $\mathbb{Q}_p^*$ is partitioned into countably many cosets of $\mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p)$. Normalize the Haar measure $\mu$ so that $\mu(\mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p))=1$. Each coset of $\mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p)$ is partitioned into $p-1$ cosets of $1+p\mathbb{Z}_p$ (each of measure $\frac{1}{p-1}$). Each coset of $1+p\mathbb{Z}_p$ is partitioned into $p$ cosets of $1+p^2\mathbb{Z}_p$ (each of measure $\frac{1}{p(p-1)}$) and so on.

This iterated partition has another nice property - From the level of cosets of $1+p\mathbb{Z}_p$ onwards, each part in each level is a $p$-adic ball, and each $p$-adic balls appears as a part in the appropriate level of the iterated partition.

That was the topology and Haar measure on $\mathbb{Q}_p^*$. Now, back to $C_c^\infty(\mathbb{Q}_p^*)$.

A function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ is compactly supported exactly when its support is contained in a finite number of cosets of $\mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p)$ (since a compact set is totally bounded and these cosets are open sets). The function $f$ is locally constant if it is a (finite) linear combination of characteristic functions of open sets.

So, the conclusion in that a function $f\in C_c(\mathbb{Q}_p^*)$ is, in general, made up in the following way:

1. Choose a finite number of complex numbers for the range: $z_0=0,z_1,\dotsc,z_m\in\mathbb{C}$.
2. Choose a finite number of cosets of $\mathbb{F}_p^*\cdot(1+p\mathbb{Z}_p)$.
3. In each of the chosen cosets, associate each part in the iterated partition of the coset with one of the $z_i$, where a part may be associated with $z_i$ for $i\neq 0$ only if all parts containing it are associated with $z_0=0$.
4. The association of values to parts in the iterated partition defines a linear combination of characteristic functions of open sets. Integrating such function is obvious by the previous discussion of the Haar Measure.

My questions are:

1. Main question: Is what I presented correct?
2. Any suggestions?
3. This was a warmup before looking at $C_c^\infty(\text{GL}_2(\mathbb{Q}_p))$. Any suggestions before going there?

Answer 1

I find point 3. of your description a little hard to parse, so I would phrase it a different way. (The discussion prior to that looked fine.)

Here is how I would phrase it:

You have to choose some $n \geq 0$, and then look at finitely many cosets of $1 +p^n\mathbb Z_p$ (or $\mathbb Z_p^{\times}$ if $n = 0$). To each of these cosets you assign a complex number.

(I think of $n$ as prescribing a radius of constancy for the given locally constant function.)