  In this article, we propose to use the character theory of compact Lie groups
and their orthogonality relations for the study of Frobenius distribution and
Sato-Tate groups. The results show the advantages of this new approach in
several aspects. With samples of Frobenius ranging in size much smaller than
the moment statistic approach, we obtain very good approximation to the
expected values of these orthogonality relations, which give useful information
about the underlying Sato-Tate groups and strong evidence of the correctness of
the generalized Sato-Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points
provide satisfactory convergence. Even for $g = 2$, the classical approach
using moment statistics requires about $2^{30}$ sample points to obtain such
information.
