$W_\infty$ – a geometric approach

A brief review is given of an adaptation of the coadjoint orbit method appropriate for the study of models with infinite-dimensional symmetry groups. It is illustrated on several examples, including derivation of the WZNW action of induced $D=2$ $(N,0)$ supergravity. As the main application, we present the geometric action on a generic coadjoint orbit of the deformed group of area-preserving diffeomorphisms. This action is precisely the anomalous effective WZNW action of $D=2$ matter fields coupled to a chiral $W_\infty$ gravity background. Similar actions are given which produce the KP hierarchy as on-shell equations of motion.