The space of mm-ary differential operators as a module over the Lie algebra of vector fields

The space Dλ¯;μ, where λ¯=(λ1,…,λm), of mm-ary differential operators acting on weighted densities is a (m+1)(m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between the space Dλ¯;μ and the corresponding space of symbols as sl(2)sl(2)-modules. This yields to the notion of the sl(2)sl(2)-equivariant symbol calculus for mm-ary differential operators. We show, however, that these two modules cannot be isomorphic as sl(2)sl(2)-modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules Dλ¯;μ2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.