Hermitian symmetric space

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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.

The irreducible compact Hermitian symmetric spaces are classified as follows.

G	H/K	complex dimension	geometric interpretation
$\mathrm{SL}(p+q,\mathbb{C})$	$\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(q))$	$p q$	Grassmannian of complex p-dimensional subspaces of $\mathbb{C}^{p+q}$
$\mathrm{SO}(p+2,\mathbb{C})$	$\mathrm{SO}(p+2)/\mathrm{SO}(p) \times \mathrm{SO}(2)$	$p$	Grassmannian of oriented real 2-dimensional subspaces of $\mathbb{R}^{p+2}$
$\mathrm{SO}(2n,\mathbb C)$	$\mathrm{SO}(2n)/\mathrm{U}(n)\,$	$\tfrac 12 n(n-1)$	Space of orthogonal complex structures on $\mathbb{R}^{2n}$
$\mathrm{Sp}(2n,\mathbb C)$	$\mathrm{Sp}(n)/\mathrm{U}(n)\,$	$\tfrac 12 n(n+1)$	Space of complex structures on $\mathbb{H}^n$ compatible with the inner product
$E_6^{\mathbb C}$	$E_6/\mathrm{SO}(10)\times\mathrm{SO}(2)$	16	Complexification of the Cayley projective plane $\mathbb{OP}^2$
$E_7^{\mathbb C}$	$E_7/E_6\times\mathrm{SO}(2)$	27

In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces are the four infinite series AIII, BDI with p = 2, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact case is similar.