The sets of naturals, signed integers, rationals, reals are denoted by the standard symbols $\mathbf{N},\mathbf{Z},\mathbf{Q},\mathbf{R}$ or $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$. Is there a similar symbol commonly used to denote the recursive (aka computable) reals?
Rice, in the paper where he proves they are a field, Recursive real numbers (1954), denotes their field with "$\mathscr{E}$".
The notation I've most commonly seen in this context is "$\mathsf{REC}$." This is also used variously to denote the set of computable binary sequences (= computable points in Cantor space) and the set of computable sequences of natural numbers (= computable points in Baire space), so the context needs to be clear enough to indicate which meaning is meant. When multiple versions are considered at once, in my experience the "context" is added as a subscript, as $\mathsf{REC}_\mathbb{R}$ vs. $\mathsf{REC}_{2^\omega}$ vs. $\mathsf{REC}_{\omega^\omega}$ respectively.
For whatever reason, I've not seen "$\mathsf{COM}$" or "$\mathsf{COMP}$" used analogously with anywhere near the same frequency.