Raf Bocklandt and Lieven

Recently, V. Ginzburg proved that Calogero phase space is a coad-joint orbit for some innnite dimensional Lie algebra coming from noncommuta-tive symplecticgeometry, 8]. In this note we generalize his argument to speciic quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg 9]. Using results of W. Crawley-Boevey and M. Holland 6], 4] and 5] we give a combinatorial description of all the relevant couples (;) which are coadjoint orbits. Finally we explain the coadjoint settings as those for which there is a Cayley-smooth algebra associated to them. 1. Introduction. Over the last couple of years some surprising new results were obtained about the space Weyl of isomorphism classes of right ideals in the rst Weyl algebra A 1 (C) linking its structure to the adelic Grassmannian Gr ad and certain moduli spaces.