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Let $\pi :Z\to X$ denote a 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 $Y_{\lambda }=Z/\mathrm{Stab}\left(\lambda \right)$. One defines a Prym variety $P_{\lambda }\subset \mathrm{Jac}\left(Y_{\lambda }\right)$ and we denote by ${\varphi }_{\lambda }$ the restriction of the principal polarization of $\mathrm{Jac}\left(Y_{\lambda }\right)$ upon $P_{\lambda }$. For two dominant weights $\lambda$ and $\mu$, we construct a correspondence $S_{\lambda \mu }$ on $Y_{\lambda }\times Y_{\mu }$ and calculate the pull-back of ${\varphi }_{\mu }$ by $S_{\lambda \mu }$ in terms of ${\varphi }_{\lambda }$.