Siegel upper half-plane

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In mathematics, a Siegel upper half-plane is the set of n×n symmetric matrices over the complex number field whose imaginary part is positive definite. The name is for Carl Ludwig Siegel.

For example, when n = 1, the Siegel upper half-plane is the upper half-plane.

There is a group action of the symplectic group

Sp_2n(R)

on the Siegel upper half-plane, where

Define