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We construct a co-$t$-structure on the derived category of coherent sheaveson the nilpotent cone $\mathcal{N}$ of a reductive group, as well as on thederived category of coherent sheaves on any parabolic Springer resolution.These structures are employed to show that the push-forwards of the "exoticparity objects" along the (classical) Springer resolution give indecomposableobjects inside the coheart of the co-$t$-structure on $\mathcal{N}$. We alsodemonstrate how the various parabolic co-$t$-structures can be related byintroducing an analogue to the usual translation functors. As an application,we give a proof of a scheme-theoretic formulation of the relative Humphreysconjecture on support varieties of tilting modules in type $A$ for $p>h$.

Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules