Several complex variables and analytic spaces Complex spaces with automorphism groups Theorem

Cartan's classification of bounded symmetric domains

Builds on Bounded symmetric domain Biholomorphism Hermitian symmetric space Cartan's classification of symmetric spaces

Statement

Every bounded symmetric domain is biholomorphic to a product of irreducible bounded symmetric domains, and the factors are unique up to order and biholomorphism.
Every irreducible bounded symmetric domain is biholomorphic to one of the following domains. Here $Z^*$ is the conjugate transpose of $Z$ , and $M > 0$ means that the Hermitian matrix $M$ is positive definite.
- type $I_{m,n}$ : $\set{Z \in M_{m \times n}(\C) : I_n - Z^* Z > 0}$ ;
- type $II_n$ : $\set{Z \in M_{n \times n}(\C) : Z^T = -Z,\ I_n - Z^* Z > 0}$ ;
- type $III_n$ : $\set{Z \in M_{n \times n}(\C) : Z^T = Z,\ I_n - Z^* Z > 0}$ ;
- type $IV_n$ : $\set{z \in \C^n : \abs{z^T z} < 1,\ 1 - 2 z^* z + \abs{z^T z}^2 > 0}$ ;
- two exceptional domains, of complex dimensions $16$ and $27$ .

Remarks

Élie Cartan obtained the classification in 1935. The proof passes to Lie theory. A bounded symmetric domain is $G/K$ , where $G$ is the identity component of its automorphism group, a semisimple Lie group, and $K$ is a maximal compact subgroup. Irreducible domains correspond to simple $G$ for which $K$ has a nontrivial centre. Reading these off from the classification of symmetric spaces gives the list: $G$ is $SU(m, n)$ , $SO^*(2n)$ , $Sp(2n, \R)$ , $SO_0(n, 2)$ , or real forms of $E_6$ and $E_7$ .

Example: $I_{m,1}$ is the unit ball of $\C^m$ , and $I_{1,1} = III_1$ is the unit disc. The type $III_n$ domain is the bounded model of the Siegel upper half-space of degree $n$ .

For small indices some types coincide, so the list is without repetitions only for suitable ranges of the indices.

Going deeper

For a student

Through the list, analysis on bounded symmetric domains reduces to matrix computations. The domains are the Hermitian symmetric spaces of noncompact type, and each has a compact dual, such as a Grassmannian for type $I$ . Quotients of these domains by arithmetic groups include the Shimura varieties and the moduli spaces of abelian varieties, so the list organises much of the theory of automorphic forms.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Biholomorphism concept
Bounded symmetric domain concept
Hermitian symmetric space concept
Holonomy concept
Riemannian manifold concept
Symmetric space concept
Cartan's classification of symmetric spaces statement
Cartan's classification of bounded symmetric domains statement

Tag	`QYVY3DV9K4`
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Kind	statement, theorem
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About	Bounded symmetric domain
Uses	Bounded symmetric domain, Biholomorphism, Hermitian symmetric space, Cartan's classification of symmetric spaces