Siegel Half-space

In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by . It is the symmetric space associated to the symplectic group .

The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case ''g=1''. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper half-space in terms of its real and imaginary parts as ''Z = X + iY'', all metrics with isometry group are proportional to
:$d s^2 = \text(Y^ dZ Y^ d \bar).$

The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure $\omega$, on the underlying $2n$ dimensional real vector space $V$, i.e. the set of $J \in Hom(V)$ such that $J^2 = -1$ and $\omega(Jv, v) > 0$ for all vectors $v \ne 0$ Bowman

See also

*Siegel domain, a generalization of the Siegel upper half space
*Siegel modular form, a type of automorphic form defined on the Siegel upper half-space
* Siegel modular variety, a moduli space constructed as a quotient of the Siegel upper half-space
*Moduli of abelian varieties

References

*.

*

*

*

Complex analysis
Automorphic forms
Differential geometry
1939 introductions

{{differential-geometry-stub