Notation in Paper on Haag-Lopuszanski-Sohnius Theorem

In section 2 of ‘All possible generators of supersymmetries of the S-matrix’, an operator $G$ on the Hilbert space of states is introduced that has to commute with the S-matrix and act additively on multiple particle states. They state that this requirements are equivalent to $G$ inducing an infinitesimal transformation on the space of fields

$Psi rightarrow Psi + epsilon delta_G Psi$

such that $delta_G(Psi(x)) = i [G,Psi(x)]_{pm}$ (sign depending on the type of operator and field) is again local. I am a mathematician, so I am not really familiar with this notation. Can someone explain what this $delta_G$ is? Is is some kind of derivation, or even an operator that acts on the space of fields? As far as I know, such an infinitesimal transformation would be given by a family of sections of the pullback bundles

$xi_{Psi} in Gamma(M;Psi^* TX)$,

but I do not see how this would come to play here.