Purity of equivalued affine Springer fibers

Fixed points and components of equivalued affine Springer fibers

For G a semisimple algebraic group, we revisit the description of the components of the affne Springer fiber given by ts, with s a regular semisimple element. We then compute the fixed points of each component of a particular affne Springer fiber for Type A

Sur l'homologie des fibres de Springer affines tronquées

Following Goresky, Kottwitz and MacPherson, we compute the homology of truncated affine Springer fibers in the unramified case but under a purity assumption. We prove this assumption in the equivalued case. The truncation parameter is viewed as a divisor on an l-adic toric variety. For each fiber, we introduce a graded quasi-coherent sheaf on the toric variety whose space of global sections is precisely the l-adic homology of the truncated affine Springer fiber. Moreover, for some families of endoscopic groups, these sheaves show up in an exact sequence. As a consequence we prove Arthur's weig…

$\mathbf{GL}_{n}$ 的各向异性仿射 Springer 纤维的纯度

Purity of the anisotropic affine Springer fibers for $\mathbf{GL}_{n}$

对于 $\mathrm{GL}_{n}$ 群和各向异性元素，证明了 Goresky-Kottwitz-MacPherson 的纯度猜想，即仿射 Springer 纤维在 Grothendieck-Deligne 意义上是上同调纯的。

Homology of affine Springer fibers in the unramified case

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral which arises in the fundamental lemma is equal to the Lefschetz trace of Frobenius acting on the etale cohomology of a related variety.

On the fundamental domain of affine Springer fibers

For $G$ a connected reductive group, $γ\in \kg(F)$ semisimple regular integral, we introduce a fundamental domain $F_γ$ for the affine Springer fibers $\xx_γ$. There is a beautiful way to reduce the purity conjecture of $\xx_γ$ to that of $F_γ$, 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 prov…

Sur le lemme fondamental pour les groupes unitaires

Assuming the purity conjecture for the affine Springer fibers which has been formulated by Goresky, Kottwitz and MacPherson, we prove a geometric analog of the fundamental lemma for unitary groups. Our approach is similar to the one of Goresky, Kottwitz and MacPherson. Our main new ingredient is the link between affine Springer fibers and compactified Jacobians which is described in math.AG/0204109.