Purity conjecture for generalized affine Springer fibers

A generalized affine Springer fiber is the variety associated with a reductive group, a finite-dimensional representation, and an element of the corresponding loop-space representation; its cohomology is considered with rational coefficients. Purity conjecture. Under mild conditions, the (co)homology of a generalized affine Springer fiber is pure. This extends the purity conjecture for ordinary affine Springer fibers, which underlies applications to orbital integrals and the fundamental lemma. The general purity conjecture for affine Springer fibers remains open, and the source does not specify the precise mild conditions or a resolution in the generalized setting.