A twisted topological trace formula for Hecke operators and liftings from symplectic to general linear groups

For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of ℚ-rank r, we construct a compact manifold $\tilde X$ by gluing together 2r copies of the Borel–Serre compactification of X. We apply the classical Lefschetz fixed point formula to $\tilde X$ and get formulas for the traces of Hecke operators ℋ acting on the cohomology of X. We allow twistings of ℋ by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G,η) . As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.