A LOOP ALGEBRA CO-ADJOINT ORBIT CONSTRUCTION OF THE GENERALIZED KDV HIERARCHIES

Using a pencil of r-matrices on the algebra g X C[[z]] we prove that t he KdV hierarchies have an Adler-Kostant-Symes construction on the und erlying current algebra C(infinity)(S1, g X C[[z]]). The coadjoint orb its are reduced by Hamiltonian symmetries. The reduction process repro duces the gauge group and the bi-Hamiltonian structure. We analyse the momentum maps of these reductions, obtaining the level sets and the l ittle group. This provides a gauge fixing. Conformal symmetry of these Kdv hierarchies is analysed, an expression for the energy momentum te nsor is obtained. Our analysis is restricted to theories with Lax oper ators linear in the loop variable z.