In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian. The complex structure is automatically preserved by the isometry group H of the metric, and so M is a homogeneous complex manifold.
Some examples are complex vector spaces and complex projective spaces, with their usual Hermitian metrics and Fubini-Study metrics, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric. The compact Hermitian symmetric spaces are projective varieties, and admit a strictly larger Lie group G of biholomorphisms with respect to which they are homogeneous: in fact, they are generalized flag manifolds, i.e., G is semisimple and the stabilizer of a point is a parabolic subgroup P of G. Among (complex) generalized flag manifolds G/P, they are characterized as those for which the nilradical of the Lie algebra of P is abelian. The non-compact Hermitian symmetric spaces can be realized as bounded domains in complex vector spaces.
Hermitian symmetric spaces are used in the construction of the holomorphic discrete series representations of semisimple Lie groups.
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