The electronic structure of the (Ge,Mn)Te diluted magnetic semiconductors was investigated theoretically from first principles, using the fully relativistic Dirac linear muffin-tin orbital (LMTO) band structure method. The electronic structure is obtained with the local spin-density approximation (LSDA) as well as the LSDA+U method. The x-ray magnetic circular dichroism (XMCD) spectra of (Ge,Mn)Te DMSs at the Mn L 2,3 edges are investigated theoretically from first principles. The origin of the XMCD spectra in the compound is examined. The calculated results are compared with available experimental data.