Identities and invariant operators on homogeneous spaces (Q5959425)

Let \(G\) be a real connected \(n\)-dimensional Lie group, let \({\mathfrak G}\) be its Lie algebra, \(M= G/H\) is an \(m\)-dimensional right \(G\)-space and let \(H\) be a closed subgroup in \(G\). The generators of the transformation group \(G\) are denoted \(X_i\), \(i= 1,\dots, n\). The first main object of the study is the algebra \(D(M)\) of invariant differential operators on \(M\), i.e. operators commuting with all generators \(X_i\). The second main object of investigation are the identities on homogeneous spaces, i.e. functional relations between the generators \(X_i\) of the transformation group \(G\) that form the algebra \({\mathfrak G}\). The relations between these two objects are investigated using the method of orbits of the coadjoint representation. A classification of the homogeneous spaces is suggested on the basis of the structure of coadjoint orbits.