  It is known from the monograph [1, Chapter 5] that the weak convergence
analysis of numerical schemes for stochastic Maxwell equations is an unsolved
problem. This paper aims to fill the gap by establishing the long-time weak
convergence analysis of the semi-implicit Euler scheme for stochastic Maxwell
equations. Based on analyzing the regularity of transformed Kolmogorov equation
associated to stochastic Maxwell equations and constructing a proper continuous
adapted auxiliary process for the semi-implicit scheme, we present the
long-time weak convergence analysis for this scheme and prove that the weak
convergence order is one, which is twice the strong convergence order. As
applications of this result, we obtain the convergence order of the numerical
invariant measure, the strong law of large numbers and central limit theorem
related to the numerical solution, and the error estimate of the multi-level
Monte Carlo estimator. As far as we know, this is the first result on the weak
convergence order for stochastic Maxwell equations.
