Translation of natural operators on manifolds with AHS-structures.

*(English)*Zbl 0881.58075The Jantzen-Zuckermann translation principle is a very important tool used in the classification of invariant (homogeneous) differential operators on homogeneous spaces \(M=G/P\) (with \(G\) simple and \(P\) a parabolic subgroup). It makes possible to construct inductively more complicated operators out of simpler ones. In the paper under review , invariant operators are studied for one special family of couples \((G,P)\), for Hermitian symmetric spaces. Such couples can be characterized by the fact that the corresponding Lie algebra \({\mathfrak g}\) is \(|1|\)-graded. Invariant operators studied in the paper are constructed in the curved case, i.e., on Cartan’s generalized spaces associated with the corresponding couple \((G,P)\) (such generalized spaces are also called manifolds with almost Hermitian symmetric structure).

Even in this much more general setting, a curved analogue of the translation principle is available. It makes possible to construct curved analogues of homogeneous invariant differential operators out of simpler ones. Different versions of curved translation exists (see papers by Bailey, Baston, Eastwood, Gover, Rice and Slovák quoted in the paper). The version presented in the paper has certain advantages – its bigger generality and its use to produce explicit formulae.

In more detail, to translate an invariant operator \(D:\Gamma(\underline W)\to\Gamma(\underline W')\) between bundles associated to irreducible \(P\)-modules \(W,W'\) on a given manifold with AHS structure to a more complicated one, it is necessary to choose a finite dimensional representation \(V\) of the group \(G\) first. Then it is possible to construct a twisted operator \(D_V\) among bundles associated to \(W\otimes V\), resp. \(W'\otimes V\). Next ingrediences needed are two translation operators. In this part of the paper, the representations \(V\) considered are restricted to the graded representation of length 2, i.e., we suppose that \(V=V_0\oplus V_1\) as a graded representation of the graded Lie algebra \({\mathfrak g}\). There are algebraic invariant operators available induced by invariant embeddings of a submodule \(\widetilde W\) of \(W\otimes V_1\) into \(W\otimes V\) and by the projection of \(W'\otimes V\) onto a submodule \(\widetilde W'\) of the quotient of \(W'\otimes V\) by \(W'\otimes V_1\). Moreover, a first order invariant operator is constructed from the bundle associated to an irreducible submodule \(\widetilde W\subset W\otimes V_0\) to the bundle associated to \(W\otimes V\). In a similar way, a first order invariant operator is constructed from the bundle associated to \(W'\otimes V\) to a bundle associated to an irreducible submodule \(\widetilde W'\) of \(W'\otimes V_1\). Having these (algebraic or differential) translation operators, new invariant operators \(D':\Gamma(\underline{\widetilde W})\to\Gamma(\underline{\widetilde W'})\) are created by composing them with the operator \(D_V\).

In the second part of the paper, interesting examples showing how the procedure works are worked out explicitly for the case of almost Grassmannian structures.

Even in this much more general setting, a curved analogue of the translation principle is available. It makes possible to construct curved analogues of homogeneous invariant differential operators out of simpler ones. Different versions of curved translation exists (see papers by Bailey, Baston, Eastwood, Gover, Rice and Slovák quoted in the paper). The version presented in the paper has certain advantages – its bigger generality and its use to produce explicit formulae.

In more detail, to translate an invariant operator \(D:\Gamma(\underline W)\to\Gamma(\underline W')\) between bundles associated to irreducible \(P\)-modules \(W,W'\) on a given manifold with AHS structure to a more complicated one, it is necessary to choose a finite dimensional representation \(V\) of the group \(G\) first. Then it is possible to construct a twisted operator \(D_V\) among bundles associated to \(W\otimes V\), resp. \(W'\otimes V\). Next ingrediences needed are two translation operators. In this part of the paper, the representations \(V\) considered are restricted to the graded representation of length 2, i.e., we suppose that \(V=V_0\oplus V_1\) as a graded representation of the graded Lie algebra \({\mathfrak g}\). There are algebraic invariant operators available induced by invariant embeddings of a submodule \(\widetilde W\) of \(W\otimes V_1\) into \(W\otimes V\) and by the projection of \(W'\otimes V\) onto a submodule \(\widetilde W'\) of the quotient of \(W'\otimes V\) by \(W'\otimes V_1\). Moreover, a first order invariant operator is constructed from the bundle associated to an irreducible submodule \(\widetilde W\subset W\otimes V_0\) to the bundle associated to \(W\otimes V\). In a similar way, a first order invariant operator is constructed from the bundle associated to \(W'\otimes V\) to a bundle associated to an irreducible submodule \(\widetilde W'\) of \(W'\otimes V_1\). Having these (algebraic or differential) translation operators, new invariant operators \(D':\Gamma(\underline{\widetilde W})\to\Gamma(\underline{\widetilde W'})\) are created by composing them with the operator \(D_V\).

In the second part of the paper, interesting examples showing how the procedure works are worked out explicitly for the case of almost Grassmannian structures.

Reviewer: V.Souček (Praha)