The cohomology of left-invariant involutive structures
Final Report Abstract
The focus of this project is the study of left-invariant involutive structures on compact Lie groups. These structures are defined by left-invariant complex subbundles of the complexified tangent bundle, which are closed under the Lie bracket and generalize the notions of complex and CR structures. They are relevant in differential geometry, representation theory, and the study of several complex variables. The central part of the work involves developing techniques to relate different notions of cohomology spaces associated with involutive structures. The key question is whether the global cohomology of the differential complex defined by the involutive structure could be computed using only left-invariant forms. While this holds for compact Lie groups in the cases of De Rham cohomology (a classical result by Chevalley and Eilenberg) and Dolbeault cohomology (a result by Pittie), it is not guaranteed in more general settings. The main results of the project include the extension of these results in two directions: the elliptic case and the CR case. In parallel, the project also explored the analysis of differential operators naturally associated with these structures. Beyond the obtained results, the report briefly mentions future research directions, such as a deeper analysis of the essentially real case and the study of global index theory for elliptic operators, aiming to expand the obtained results. In summary, the project deepened the understanding of the relationship between left-invariant involutive structures and global analysis on Lie groups, providing both foundational results and tools for future research in differential geometry, Lie theory, and the analysis of partial differential equations.
Publications
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Levi-flat CR structures on compact Lie groups. Annals of Global Analysis and Geometry, 64(1).
Jacobowitz, Howard & Jahnke, Max Reinhold
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Global solvability and cohomology of tube structures on compact manifolds. Mathematische Annalen, 390(2), 2199-2233.
Araújo, Gabriel; Ferra, Igor A.; Jahnke, Max R. & Ragognette, Luis F.
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A class of globally analytic hypoelliptic operators on compact Lie groups. The Journal of Geometric Analysis, 35(9).
Jahnke, Max Reinhold & Braun Rodrigues, Nicholas
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Closed elliptic structures on compact semisimple Lie groups. Proceedings of the American Mathematical Society, 154(5), 2221-2234.
Jahnke, Max Reinhold
