On the fundamental domain of affine Springer fibers

Abstract

For $G$ a connected reductive group, $γ\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_{\gamma }$ for the affine Springer fibers $\xx_γ$. There is a beautiful way to reduce the purity conjecture of $\xx_γ$ to that of $F_{\gamma }$, we call it the Arthur-Kottwitz reduction. When restricted to the unramified case, it turns out that these fundamental domains behave well in family. We formulate a rationality conjecture about a generating series of their Poincaré polynomials. We then study them in detail for the group $\gl_{3}$. In particular, we pave them in affine spaces and we prove the rationality conjecture.

Final version, to appear in Math. Z

References in corpus (2)

• Pureté des fibres de Springer affines pour $G{L}_4$
• Truncated affine grassmannians and truncated affine Springer fibers for ${\mathrm{GL}}_3$