  Two major results in the theory of l-adic mixed constructible sheaves are the
purity theorem (every simple perverse sheaf is pure) and the decomposition
theorem (every pure object in the derived category is a direct sum of shifts of
simple perverse sheaves). In this paper, we prove analogues of these results
for coherent sheaves. Specificially, we work with staggered sheaves, which form
the heart of a certain t-structure on the derived category of equivariant
coherent sheaves. We prove, under some reasonable hypotheses, that every simple
staggered sheaf is pure, and that every pure complex of coherent sheaves is a
direct sum of shifts of simple staggered sheaves.
