Compact Hermitian Symmetric Spaces
The irreducible compact Hermitian symmetric spaces H/K are classified as follows.
| G | H | K | complex dimension | geometric interpretation |
|---|---|---|---|---|
| pq | Grassmannian of complex p-dimensional subspaces of | |||
| p | Grassmannian of oriented real 2-dimensional subspaces of | |||
| Space of orthogonal complex structures on | ||||
| Space of complex structures on compatible with the inner product | ||||
| 16 | Complexification of the Cayley projective plane | |||
| 27 | Space of symmetric submanifolds of Rosenfeld projective plane which are isomorphic to |
In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces are the four infinite series AIII, BDI with p = 2 or q = 2, DIII and CI, and two exceptional spaces, namely EIII and EVII.
The realization of H/K as a generalized flag variety G/P is obtained by taking G as in the table (a complexification of H) and P equal to the semidirect product of L with the complexified isotropy representation of K, where L (the Levi factor of P) is the complexification of K.
At the Lie algebra level, there is a symmetric decomposition
where is a real vector space with a complex structure J, whose complex dimension is given in the table. Correspondingly, there is a graded Lie algebra decomposition
where is the decomposition into +i and −i eigenspaces of J and . The Lie algebra of P is the semidirect product . It follows that the exponential image of modulo P realizes the complex vector space as a dense open subset of G/P.
Read more about this topic: Hermitian Symmetric Space
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