Arithmetically Cohen--Macaulay bundles on homogeneous varieties of Picard rank one

In this paper, we study arithmetically Cohen–Macaulay (ACM) bundles on homogeneous varieties $G\mathrm{/}P$. Indeed we characterize the homogeneous ACM bundles on $G\mathrm{/}P$ of Picard rank one in terms of highest weights. This is a generalization of the result on $G\mathrm{/}P$ of classical types presented by Costa and Miró-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a consequence we prove that only finitely many irreducible homogeneous ACM bundles, up to twisting line bundles, exist over all such $G\mathrm{/}P$. Moreover, we derive the list of the highest weights of the irreducible homogeneous ACM bundles on particular homogeneous varieties of exceptional types such as the Cayley Plane and the Freudenthal variety.