I'm sorry if this question is awkwardly phrased -- I'm very much an amateur at algebraic geometry, but this question came up in my research.
Here goes. Let $\mathbb{H}^g$ be the genus $g$ Siegel upper half plane. Thus the symplectic group $Sp_{2g}(\mathbb{Z})$ acts on $\mathbb{H}^g$ with quotient the moduli space of principally polarized abelian varieties (PPAV's). Let $\Gamma < Sp_{2g}(\mathbb{Z})$ be a finite subgroup. Define $X(\Gamma) \subset \mathbb{H}^g$ to be the set of all $p$ such that $\gamma(p)=p$ for all $\gamma \in \Gamma$. The set $X(\Gamma)$ is then an analytic subvariety and $\Gamma$ acts as a group of automorphisms of the PPAV's corresponding to the points of $X(\Gamma)$.
Recall that a PPAV is simple if it doesn't split as a nontrivial direct sum of PPAV's. I'm interested in finite subgroups $\Gamma < Sp_{2g}(\mathbb{Z})$ such that none of the PPAV's corresponding to points of $X(\Gamma)$ are simple. A dream would be to have a classification of such subgroups, but even interesting examples would be very helpful.
