Semiholonomic Verma modules
Although the motivation for this paper comes from conformal differential geometry, its main part is purely algebraic. For a Lie group \(G\) and a closed Lie subgroup \(P\) of \(G\) the authors first define a semiholonomic version of \(\overline{\mathcal U}(\mathfrak g)\) of the universal enveloping algebra of the Lie algebra \(\mathfrak g\) of \(G\). Using this algebra, they define for a finite-dimensional representation \(\mathbb E\) of \(P\) a semiholonomic induced module \(\overline V(\mathbb E)\) which is related to semiholonomic jets of sections of the homogeneous bundle \(G\times_{P}\mathbb E\) in the same way as the usual induced module is related to usual (holonomic) jets. They prove that for such modules there is an analog of Frobenius reciprocity and that the usual induced module is naturally a quotient of the semiholonomic one. The main part of the paper answers the question of which homomorphisms between induced modules lift to homomorphisms between their semiholonomic counterparts in the case where \(G=\text{Spin}_{0}(n+1,1)\) and \(P\) is the maximal parabolic subgroup for which \(G/P\) is the conformal sphere \(S^n\). They recall the known description of all homomorphisms of induced modules (which are just the generalized Verma modules in this case), and show that all homomorphisms except one family of nonstandard nonsingular homomorphisms for even \(n\) lift to the semiholonomic induced modules. For this exceptional family, it is shown that the homomorphisms do not lift. The proof is modeled on the Jantzen-Zuckermann translation principle for homomorphisms of generalized Verma modules. In an appendix, the authors discuss the application of their results to conformal differential geometry, obtaining a conceptual and clear proof of the known results on existence of curved analogs on arbitrary conformal manifolds of conformally invariant differential operators on the sphere.
conformally invariant differential operator
scientific article; zbMATH DE number 1104107
scientific article