Prym Subvarieties of Jacobians via Schur correspondances between curves

Let $\pi : Z \to X$ be Galois cover of smooth projective curves with Galois group $W$ a Weyl group of a simple Lie group $G$. For a dominant weight $\lambda$, we consider the intermediate curve $Y_\lambda= Z/\Stab(\lambda)$. One can realise a Prym variety $P_\lambda \subset \Jac(Y_\lambda)$ and we denote $\varphi_\lambda$ the restriction of the principal polarisation of $\Jac(Y_\lambda)$ upon $P_\lambda$. For two dominant weights $\lambda$ and $\mu$, we construct a correspondence $\Delta_{\lambda \mu}$ on $Y_\lambda \times Y_\mu$ and calculate the pull-back of $\varphi_\mu$ by $\Delta_{\lambda \mu}$ in terms of $\varphi_\lambda$.