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, On the dependence of the affine Springer fibers on the root valuation datum
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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.