Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions

For any $nge 2$ we study the group algebra decomposition of an $([frac{n}{2}]+1)$ -dimensional family of principally polarized abelian varieties of dimension $n$ with an action of the dihedral group of order $2n$ . For any odd prime $p, n=p$ and $n=2p$ we compute the induced polarization on the isotypical components of these varieties and some other distinguished subvarieties. In the case of $n=p$ the family contains a one-dimensional family of Jacobians. We use this to compute a period matrix for Klein’s icosahedral curve of genus 5.