engleski

Singular BGG complexes for the symplectic case

Let G be a semisimple Lie group and P its parabolic subgroup. It is well known that any finite-dimensional simple G-module allows a resolution by invariant differential operators acting between direct sums of homogeneous bundles over the generalized flag manifold G/P. Such a resolution is called the Bernstein- Gelfand-Gelfand (BGG for short) resolution. In the dual setting, this corresponds to the resolution of a finite-dimensional simple g- module by direct sums of generalized Verma modules, which is also called the BGG resolution. Modules in the resolution have a regular infinitesimal character. Using the Penrose transform, we construct analogues of such resolutions in certain singular infinitesimal characters, in the holomorphic geometric setting, for type C. We take G to be the symplectic group, P its |1|-graded parabolic subgroup, so that G/P is the Lagrangian Grassmannian. We explicitly describe the operators in the resolution, and determine their order. We prove the exactness of the constructed complex over the big affine cell.

BGG ; Resolution ; Homogeneous bundles ; Invariant differential operators ; Generalized Verma modules ; Symplectic group, Lagrangian Grassmannian ; Penrose transform

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