Semisimple principal fibrations are a certain class of holomorphic fibrations Y over a product of constant scalar curvature K\"ahler manifold with fiber a compact K\"ahler manifold X. One of the main assets of these fibrations is that they come equipped with a connection which allows defining, from any Kähler metrics on X, a K\"ahler metric on Y,  called compatible metric. A K\"ahler class containing a compatible metric is said to be compatible. In this talk, after giving details of the notions above, I will explain how to translate the Calabi problem on a compatible K\"ahler class on Y, to a weighted cscK problem (in the sense of Lahdili) on the corresponding fiber X.

This is a joint work with V. Apostolov and A. Lahdili.