It seems, there are two definitions of the Siegel upper half space. 


1) One used by, say, Krantz in "Explorations in Harmonic Analysis; Page 252, or, by So Chin Chew and Mei Chi Shaw in "Partial Differential Equations in Several Variables": Page 171:


$\mathbb{H}\_{n}=\left\{ \left(z\_{1},z\_{2},\cdots,z\_{n}\right)\in\mathbb{C}^{n}\mid\mathrm{im}(z\_{1})>\sum\_{j=2}^{n}|z\_{j}|^{2}\right\} $


2) And,the other used more classically in most literature concerning Siegel Modular forms: 


$\mathbb{H}\_{g}=\left\{ Z=X+iY\in\mathrm{M\_{g}(\mathbb{C})\mid Z}\mbox{ is symmetric and }Y\mbox{ is positive definite}\right\} $


Are these two definitions equivalent, via say, some biholomorphic map connecting the two for $n=\frac{g(g+1)}{2}$? 



Connection between the two definitions of Siegel Upper Half Space
These are two different "classical domains". An on-line discussion of the four families of classical domains, related semi-simple real Lie groups, and some related matters is at <http://www.math.umn.edu/~garrett/m/lie/> in the note "classical groups, domains, cones".


