Hermitian Symmetric Space - Compact Hermitian Symmetric Spaces

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.

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