The characterization of distinguished representations is crucial for studying automorphic representations. The celebrated conjectures of Sakellaridis and Venkatesh provide such a characterization in many cases. In particular, they provide a conjectural description of the representations of a split reductive group that are distinguished by a split reductive spherical subgroup. However, there remain many mysteries when the generic stabilizer is disconnected.

The comparison of relative trace formulae, initially suggested by Jacquet, has been one of the most effective ways to study distinction problems in automorphic representation theory. Stabilization is a pivotal step for the comparison of relative trace formulae. To prepare for stabilization, one needs to investigate the endoscopic relative orbital integrals.

In this talk, we study the endoscopy theory for unitary groups in a relative setting where the generic stabilizer is disconnected and finite over a p-adic field. This talk aims to compute an explicit formula for  endoscopic relative orbital integrals.