On the cohomological purity of the affine Springer fibers

Abstract

We address questions posed by G\'erard Laumon and Jean-Loup Waldspurger concerning the cohomological purity of affine Springer fibers. More precisely, we show that an affine Springer fiber is cohomologically pure if and only if its $\xi$-stable quotient is cohomologically pure, and that this is further equivalent to the cohomological purity of a certain sequence of truncated affine Springer fibers. We deduce from this a sheaf-theoretic reformulation of cohomological purity for affine Springer fibers. We then compare this new criterion with a previously known one via a microlocal analysis of the relevant intersection complexes. As a corollary, we show that both the primitive part of the cohomology of an affine Springer fiber and the cohomology of its $\xi$-stable quotient depend only on the root valuation datum of the defining element.