The physical subspace of the SL (2, R) current algebra based model of induced 2d gravity is investigated. Using an appropriate ground state, an equation guaranteeing the non-negativity of this subspace is derived using a BRS analysis. In case of the ground state allowing the derivation of simple differential equations for the correlation functions the presence of an SL (2, R) background charge is exhibited. It is shown that changing one ground state for the other converts the KPZ equation for the anomalous dimensions to the previous equation for physical states.