On the cohomological purity of the affine Springer fibers

TL;DR

This paper investigates the conditions under which affine Springer fibers are cohomologically pure, establishing equivalences and reformulations that connect stability, truncation, and root valuation data.

Contribution

It provides a new sheaf-theoretic criterion for cohomological purity of affine Springer fibers and compares it with existing criteria via microlocal analysis.

Findings

Cohomological purity of affine Springer fibers is equivalent to that of their $\xi$-stable quotients.

A sheaf-theoretic reformulation of cohomological purity is established.

The primitive cohomology and the cohomology of the $\xi$-stable quotient depend only on root valuation data.

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.