  Strong convergence rates for (temporal, spatial, and noise) numerical
approximations of semilinear stochastic evolution equations (SEEs) with smooth
and regular nonlinearities are well understood in the scientific literature.
Weak convergence rates for numerical approximations of such SEEs have been
investigated since about 11 years and are far away from being well understood:
roughly speaking, no essentially sharp weak convergence rates are known for
parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3
in [A. Debussche, Weak approximation of stochastic partial differential
equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for
details. In this article we solve the weak convergence problem emerged from
Debussche's article in the case of spectral Galerkin approximations and
establish essentially sharp weak convergence rates for spatial spectral
Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient
functions. Our solution to the weak convergence problem does not use Malliavin
calculus. Rather, key ingredients in our solution to the weak convergence
problem emerged from Debussche's article are the use of appropriately modified
versions of the spatial Galerkin approximation processes and applications of a
mild It\^{o} type formula for solutions and numerical approximations of
semilinear SEEs. This article solves the weak convergence problem emerged from
Debussche's article merely in the case of spatial spectral Galerkin
approximations instead of other more complicated numerical approximations. Our
method of proof extends, however, to a number of other kind of spatial,
temporal, and noise numerical approximations for semilinear SEEs.
