An affine Springer fiber is cohomologically pure if and only if its ξ-stable quotient and truncated versions are, with primitive cohomology depending only on the root valuation datum of the defining element.
Chen, A decomposition theorem for the affine Springer fibers
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On the cohomological purity of the affine Springer fibers
An affine Springer fiber is cohomologically pure if and only if its ξ-stable quotient and truncated versions are, with primitive cohomology depending only on the root valuation datum of the defining element.