Irreducible representations of Poisson-Lie superalgebras, and invariant differential operators
Let the field k be either \({\mathbb{R}}\) or \({\mathbb{C}}\). We let \(\hat {\mathcal H}(M)\) denote the Lie superalgebra formed by the functions defined on a symplectic supermanifold \((M,\omega)\), dim M\(=(2n,m)\) relative to the Poisson bracket. Let the vector bundle \(E\to M\), having a finite- dimensional fiber V, be associated with the bundle of symplectic frames on M. Then the Hamiltonian vector fields on M act on the corresponding space of sections \(\Gamma\) (E). Now if \(L\to E\) is a linear bundle endowed with a connection \(\nabla\) whose curvature form is \(h\omega\), \(h\in k\), then \(\hat {\mathcal H}(M)\) acts in the space of sections of the bundle \(E\otimes L\) according to the rule \(f\to X_ f=L_ f\otimes id+id\otimes(\nabla +hf)\), where \(L_ f\) is the Hamiltonian vector field of the function f relative to the form \(\omega\). The aim of this note is to describe the differential operators c: \(\Gamma\) \((E{}_ 1\otimes L)\to \Gamma(E_ 2\otimes L)\) that commute with this action, under the assumption that the fibers of the bundles \(E_ 1\) and \(E_ 2\) are simple, finite-dimensional osp(m,2n)-modules.
scientific article; zbMATH DE number 3860034