Equivariant Ulrich bundles on exceptional homogeneous varieties

We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6/P_1$ and the $E_7$-adjoint variety $E_7/P_1$. Moreover, we compute that $\mathcal E_{\omega_5+3\omega_6}$ is the only irreducible equivariant Ulrich bundle on the Cayley plane $E_6/P_1$, and $\mathcal E_{\omega_5+3\omega_6+8\omega_7}$ is the unique irreducible equivariant Ulrich bundle on the adjoint variety $E_7/P_1$. From this result, we see that a general hyperplane section $F_4/P_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.