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stub for parabolic geometry to record some references
Can anyone explain what’s important about parabolic geometries as kinds of Cartan geometries?
I am still picking this up myself, but a key point is that in the case of flag manifolds $G/P$ there is an important construction/result called the BGG resolution, and the big insight of Čap-Slovák-Souček 01 is that this generalizes to the curved (G/P-Cartan) case at all and thus provides even richer structure than the original BGG resolution does.
The introduction of Calderbank-Diemer 00 sums this up as follows:
Hence we now know that all standard homomorphisms of parabolic Verma modules induce differential operators also in the curved setting, providing us with a huge supply of invariant linear differential operators.
Cap emphasises the connection between parabolic geometries and underlying structures:
In a series of pioneering papers in the 1960’s and 70’s culminating in [29], N. Tanaka showed that for all semisimple Lie groups and parabolic subgroups normal Cartan geometries are determined by underlying structures. These results have been put into the more general context of filtered manifolds in the work of T. Morimoto (see e.g. [23]) and a new version of the result tailored to the parabolic case was given in [12].
Added some more scraps, including link to CR structures which needs to be written. Will there be ’higher parabolic geometries’?
have added to various related entries (parabolic geometry, conformal geometry, CR geometry) pointers to
which is concise and informative. More later…
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