Sur l'homologie des fibres de Springer affines tronquées

奇异支集与仿射 Springer 纤维的同调

Support singulier et homologie des fibres de Springer affines

本文构建无穷维叠的奇异支集理论，并用其计算 Grothendieck–Springer 仿射 Springer 层的奇异支集，从而关联 Goresky–Kottwitz–McPherson 的局部常定性猜想。

Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical cas…

Poincare polynomials of character varieties, Macdonald polynomials and affine Springer fibers

We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_λ[X;q,t]$ is i…

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.

The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.

"Del Pezzo surfaces as Springer fibres for exceptional groups"

We show that simultaneous log resolutions of simply elliptic singularities can be constructed inside suitable stacks of principal bundles over elliptic curves. In particular, we give a direct geometrical construction of del Pezzo surfaces from the corresponding exceptional simple algebraic groups.