Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

We consider Picard surfaces, locally symmetric varieties $S_{\mathrm{\Gamma }}$ attached to the Lie group SU(2,1), and we construct explicit differential forms on $S_{\mathrm{\Gamma }}$ representing Eisenstein classes, i.e. cohomology classes restricting non-trivially to the boundary of the Borel-Serre compactification. This is needed for the computation of the class of the extensions of the Hodge structure that we have constructed in [2] according to the predictions of the Bloch-Beilinson conjectures. The tool for the construction of the differential forms is an analysis of relative Lie algebra cohomology of the principal series of SU(2,1) using recent methods of Buttcane and Miller.