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Some differential complexes within and beyond parabolic geometry. (English) Zbl 1453.53014

Shoda, Toshihiro (ed.) et al., Differential geometry and Tanaka theory. Differential system and hypersurface theory. Proceedings of the international conference, RIMS, Kyoto University, Japan, January, 24–28, 2011. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 82, 13-40 (2019).
A. Čap et al. [Ann. Math. (2) 154, No. 1, 97–113 (2001; Zbl 1159.58309)] constructed sequences of invariant differential operators on parabolic geometries of any type \(G/P\), one for each finite-dimensional representation \(V\) of \(G\), where \(G\) is a semisimple Lie group and \(P\subset G\) a parabolic subgroup. These sequences are known as Bernstein-Gelfand-Gelfand (BGG) sequences. D. M. J. Calderbank and T. Diemer [J. Reine Angew. Math. 537, 67–103 (2001; Zbl 0985.58002)] simplified the construction of BGG sequences.
In the paper under review, the authors present some examples constructed in a more elementary way. Their method extends to certain non-parabolic geometries, namely arbitrary contact and symplectic geometries. The spectral sequence of a filtered complex [T. Y. Chow, Notices Am. Math. Soc. 53, No. 1, 15–19 (2006; Zbl 1092.55014)] is used as a replacement for tedious diagram chasing.
For the entire collection see [Zbl 1437.53042].

MSC:

53A40 Other special differential geometries
53D10 Contact manifolds (general theory)
58A12 de Rham theory in global analysis
58A17 Pfaffian systems
58J10 Differential complexes
58J70 Invariance and symmetry properties for PDEs on manifolds
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References:

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