Quantization of Nilpotent Coadjoint Orbits Quantization of Nilpotent Coadjoint Orbits Quantization of Nilpotent Coadjoint Orbits

Let G be a complex reductive group. We study the problem of associating Dixmier algebras to nilpotent (co)adjoint orbits of G, or, more generally, to orbit data for G. If g = 0 + n + in is a triangular decomposition of g and 0 is a nilpotent orbit, we consider the irreducible components of 0 n n, which are Lagrangian subvarieties of 0. The main idea is to construct, starting with certain "good" components of 0 n n, a Dixmier algebra which should be associated to some cover of 0. We carry out the construction if the orbit 0 is small. Then we apply this result to certain simple groups and obtain the Dixmier algebras associated to a variety of nilpotent orbits. A particularly interesting example is a non-commutative orbit datum which we call the Clifford orbit datum. By modifying our main construction a bit we obtain a Dixmier algebra which should be associated to that datum. Thesis Supervisor: David A. Vogan Title: Professor of Mathematics