  We propose two new approaches to the Tannakian Galois groups of holonomic
D-modules on abelian varieties. The first is an interpretation in terms of
principal bundles given by the Fourier-Mukai transform, which shows that they
are almost connected. The second constructs a microlocalization functor
relating characteristic cycles to Weyl group orbits of weights. This explains
the ubiquity of minuscule representations, and we illustrate it with a Torelli
theorem and with a bound for decompositions of a given subvariety as a sum of
subvarieties. The appendix sketches a twistor variant that may be useful for
D-modules not coming from Hodge theory.
