Relative Hitchin fibrations and endoscopy for Unitary Friedberg--Jacquet varieties

We prove a relative fundamental lemma for the unitary symmetric variety $G{L}_n(k^{\mathrm{\prime }})\mathrm{\backslash }{U}_{2n}$, proving a conjecture of Leslie, Xiao, and Zhang. Our proof reduces to the Lie algebra and argues over function fields in positive characteristic, using methods from Ngo-Laumon and Ngo. In particular, we reinterpret orbital integrals in terms of the cohomology of relative affine Springer fibers and globalize this using unitary versions of Hitchin moduli spaces for symmetric pairs via a product formula. As the main global input, we prove a matching of perverse sheaves between relative Hitchin moduli spaces and their corresponding endoscopic variants.