# Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane

Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i$. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix $Ae^{-R}A^T+Be^{R}B^T$ can be written as $XX^*$ where $X=Ae^{-R}+iBe^{R}$ and $X^*$ denotes the conjugate transpose of X. Is there a way to express the eigenvalues of $X$ in terms of eigenvalues of $U$?

(Is there a better way of calculating $\det(Im(\gamma Z))$ where $\gamma=\big(\begin{smallmatrix}A&B\\-B&A\end{smallmatrix}\big)\in Sp_{2n}(\mathbb{R})\bigcap SO_{2n}(\mathbb{R})$ ,and $Z=\big(\begin{smallmatrix}e^R&0\\0&e^{-R}\end{smallmatrix}\big)\cdot i \in \mathbb{H}_n$, the Siegel Upper Half Plane of degree $n$.)

• It is a bit difficult for me to understand the genuine question here. There is the old identity $\det \Im \gamma z=|\det (cz+d)|^{-2}\cdot \det \Im z$ which applies as well in the Siegel modular case as in the elliptic modular case. If the question is about better algorithms, I have no idea. – paul garrett Apr 27 '16 at 23:24