I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space $\mathscr{S}(G)$ of complex functions over a (Hausdorff) locally compact Abelian group $G$ is defined based on the notion of differential operators on groups. I have trouble understanding how differential operators are defined in groups that are discrete; see questions 1 and 2 below.

For simplicity I only look at groups of the form For simplicity (and as in the previous post ) I will always assume that $G$ is a group of the form $$G=\mathbb{R}^a\times\mathbb{T}^b\times\mathbb{Z}^c\times F$$, where $\mathbb{T}^m$ is an $m$-dimensional torus and $F$ an finite Abelian group given in the form $$F=\mathbb{Z}_{d_1}\times\cdots\times \mathbb{Z}_{d_e}.$$

For groups of the form $G$, the space $\mathscr{S}(G)$ is defined as follows: a function $f$ is in $\mathscr{S}(G)$ if $f$ is infinitely differentiable, and if $P(\partial)f\in L^2(G)$ for every polynomial (in the $\mathbb{R}^a\times \mathbb{Z}^c$ variables) differential operator $P(\partial)$ on $G$.

Question 1What is a ``polynomial differential operator $P(\partial)$'' over a group $G$? I am vaguely aware that groups of the form $G$ are the Abelian Lie groups that are compactly-generated (cf. [3]). Therefore, I assume there must exist a robust-notion of differentiability of functions over these groups. Still I do not know how differential operators on function-spaces over discrete groups such as $\mathbb{Z}^c$ or $F$ should be defined.

Question 2Why does $P(\partial)$ have to be a polynomial differential operatoronlyin the $\mathbb{R}^a\times \mathbb{Z}^c$ variables. I do not understand why $\mathbb{T}^b$ and $F$ are not mentioned. Does it perhaps something to do with compactness?

**Related posts.**

I have opened another post asking questions about an alternative equivalent way to define the Schwartz-Bruhat space via functions of rapid decay. Also, my first question seems to be related to this question.