  In [8] the authors introduced a pair of new de Rham complexes on a compact
oriented Riemannian manifold with boundary by using a pair of new boundary
conditions to discuss the refined analytic torsion on a compact manifold with
boundary. In this paper we discuss the Lefschetz fixed point formula on these
complexes with respect to a smooth map having simple fixed points and
satisfying some special condition near the boundary. For this purpose we are
going to use the heat kernel method for the Lefschetz fixed point formula.
