Siegel domain
In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel (1939). They were introduced by Piatetski-Shapiro (1959, 1969) in his study of bounded homogeneous domains.
A Siegel domain of the first kind (or first type, or genus 1) is the open subset of Cm of elements z such that
where V is an open convex cone in Rm. These are special cases of tube domains. An example is the Siegel upper half plane, where V⊂Rk(k + 1)/2 is the cone of positive definite quadratic forms in Rk and m = k(k + 1)/2.
A Siegel domain of the second kind (or second type, or genus 2), also called a Piatetski-Shapiro domain, is the open subset of Cm×Cn of elements (z,w) such that
where V is an open convex cone in Rm and F is a V-valued Hermitian form on Cn. If n = 0 this is a Siegel domain of the first kind.
A Siegel domain of the third kind (or third type, or genus 3) is the open subset of Cm×Cn×Ck of elements (z,w,t) such that
where V is an open convex cone in Rm and Lt is a V-valued semi-Hermitian form on Cn.
A bounded domain is an open connected bounded subset of a complex affine space. It is called homogeneous if its group of automorphisms acts transitively, and is called symmetric if for every point there is an automorphism acting as –1 on the tangent space. Bounded symmetric domains are homogeneous.
Élie Cartan classified the homogeneous bounded domains in dimension at most 3 (up to isomorphism), showing that they are all Hermitian symmetric spaces. There is 1 in dimension 1 (the unit ball), two in dimension 2 (the product of two 1-dimensional complex balls or a 2-dimensional complex ball). He asked whether all bounded homogeneous domains are symmetric. Piatetski-Shapiro (1959, 1959b) answered Cartan's question by finding a Siegel domain of type 2 in 4 dimensions that is homogeneous and biholomorphic to a bounded domain but not symmetric. In dimensions at least 7 there are infinite families of homogeneous bounded domains that are not symmetric.
...
Wikipedia