Bounded symmetric domain

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. With its Bergman metric, a bounded symmetric domain is a Hermitian symmetric space 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.