An integral operator into Dolbeault cohomology. (English) Zbl 0849.22014
The authors present a construction for an integral transform from sections of a vector bundle over a given manifold into Dolbeault cohomology of a related holomorphic vector bundle over another manifold. If \(Y @> \rho>> X\), \(Y @>\pi>> Z\) is a double fibration of smooth manifolds with \(Z\) having a complex structure, then the integral transform \({\mathcal S}\): \(\Gamma(X, V) \to H^S(Z, {\mathcal O}(E))\) is obtained for a suitable vector bundle \(V\) on \(X\) and a holomorphic vector bundle \(E\) on \(Z\). In the case of appropriate homogeneous situations the transform \(\mathcal S\) coincides with a certain representation theoretic intertwining operator.
Reviewer: V.Oproiu (Iaşi)
MSC:
22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
22E46 | Semisimple Lie groups and their representations |
32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |
44A12 | Radon transform |
53D50 | Geometric quantization |