  This paper deals with the weak error estimates of the exponential Euler
method for semi-linear stochastic partial differential equations (SPDEs). A
weak error representation formula is first derived for the exponential
integrator scheme in the context of truncated SPDEs. The obtained formula that
enjoys the absence of the irregular term involved with the unbounded operator
is then applied to a parabolic SPDE. Under certain mild assumptions on the
nonlinearity, we treat a full discretization based on the spectral Galerkin
spatial approximation and provide an easy weak error analysis, which does not
rely on Malliavin calculus.
