The space of [formula omitted]-ary differential operators as a module over the Lie algebra of vector fields

Abstract

The space D λ ¯ ; μ , where λ ¯ = ( λ 1 , … , λ m ) , of m -ary differential operators acting on weighted densities is a ( 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 D λ ¯ ; μ and the corresponding space of symbols as s l ( 2 ) -modules. This yields to the notion of the s l ( 2 ) -equivariant symbol calculus for m -ary differential operators. We show, however, that these two modules cannot be isomorphic as s l ( 2 ) -modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules D λ ¯ ; μ 2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.