Casimir operators on pseudodifferential operators of several variables

We study the actions of the Lie algebra nsl(2, C of SL(2, C)(n) and the associated Casimir operator on the space of pseudodifferential operators of n variables. We describe the effect of the Casimir operator on a pseudodifferential operator in connection with the symbol map and construct an nsl(2, C)-invariant lifting of the symbol map.