---
tag: Y7Q2SNKG0C
kind: concept
name: Bounded symmetric domain
msc: [32M15, 53C35]
specializes: [NT4797Z9H8]
sources:
  - url: https://en.wikipedia.org/wiki/Hermitian_symmetric_space
---

## Summary

A bounded symmetric domain is a bounded domain $D$ in $\C^n$ with many symmetries.
At each point $p$ of $D$ there is a holomorphic involution of $D$ with $p$ as an isolated fixed point.
Examples are the unit ball, the polydisc and the bounded model of the [[Siegel upper half-space|5PTYE97PF6]].
With its Bergman metric, a bounded symmetric domain is a [[Hermitian symmetric space|NT4797Z9H8]] of noncompact type, and every such space arises in this way.
Cartan classified them, and they are the domains on which automorphic forms in several variables live.

## Explanations

### L1 human

The unit disc has a holomorphic "point reflection" $z \mapsto -z$ through its centre, and by moving points around with Möbius maps it has one through every point.
A bounded symmetric domain is a bounded region in several complex variables with the same property.
These reflections are so many that they move any point to any other, so the domain looks the same everywhere.
Products of discs and the unit ball in $\C^n$ are basic examples.