  Given the nilpotent cone of a complex reductive Lie algebra, we consider its
equivariant constructible derived category of sheaves with coefficients in an
arbitrary field. This category and its subcategory of perverse sheaves play an
important role in Springer theory and the theory of character sheaves. We show
that the composition of the Fourier--Sato transform on the Lie algebra followed
by restriction to the nilpotent cone restricts to an autoequivalence of the
derived category of the nilpotent cone. In the case of $GL_n$, we show that
this autoequivalence can be regarded as a geometric version of Ringel duality
for the Schur algebra.
