Set of complex matrices with positive definite imaginary part
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 Siegel (1939). It is the symmetric space associated to the symplectic group Sp(2"g", R).
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 Sp(2"g", R). 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 SL(2, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2"g", R). 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 Sp(2"g", R) are proportional to
formula_1
The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure formula_2, on the underlying formula_3 dimensional real vector space formula_4, that is, the set of formula_5 such that formula_6 and formula_7 for all vectors formula_8.
As a symmetric space of non-compact type, the Siegel upper half space formula_9 is the quotient
formula_10
where we used that formula_11 is the maximal torus.
Since the isometry group of a symmetric space formula_12 is formula_13, we recover that the isometry group of formula_9 is formula_15. An isometry acts via a generalized Möbius transformation
formula_16
The quotient space formula_17 is the moduli space of principally polarized abelian varieties of dimension formula_18.
References.
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