I'm unsure why a command and control policy approach to pollution will still have a dead weight loss? What happens with it/ or what does the government do about it if anything?
As far as I know, there is a Least Cost Principle regarding cutting pollution, which states that: when there is a pollution target which firms (with heterogeneous and publicly unobserved pollution abatement cost curve) need to abate some pollution to achieve, the policy should aim at achieving the target with the minimum total abatement cost. Intuitively, the sufficient condition is that the marginal abatement cost (MAC) is equalised across firms (if not, it is possible to save resources, by having the firm with higher MAC to reduce a unit of abatement and let another firm of lower MAC to conduct that unit of abatement). One way to achieve this efficient outcome is a competitive tradable permit market, where the number of permits is set to the pollution target.
An CAC approach can, for example, command every firm to reduce pollution by a same amount. However, this does not guarantee the MAC is equal across firms because abatement cost curve is different for different firms. CAC cannot design an optimal plan because the cost curves are only privately observed. This yields a DWL if the benchmark is the Least Cost Principle.