Homology of affine Springer fibers in the unramified case

By Mark Goresky, R. Kottwitz, Robert D. MacPherson. 2003

105 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c1434eabfcb1e2f1f8784b530955b15bd0206f2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Thomas Lam
2. Mark Shimozono
3. Anne Schilling
4. Zhiwei Yun
5. Jennifer Morse
6. Luc Lapointe
7. Mike Zabrocki
8. Megumi Harada

In Mathematical Physics

1. Zhiwei Yun
2. Thomas Lam
3. Megumi Harada
4. Tamás Hausel
5. N. Châu
6. Mark Shimozono
7. Mike Zabrocki
8. B. Ngô

Citations in the corpus

• Fibration de Hitchin et endoscopie (2006, background): “Dans la démonstration récente du lemme fondamental pour les groupes unitaires [20], cette description du support a joué le rôle d’un énoncé de pureté analogue à une conjecture de Goresky, Kottwitz et MacPherson [9].”
• A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, background): “This influential theory and its many consequences have been extensively generalized and used since [4, 13, 15, 17, 19, 20, 21, 23, 25].”
• Global topology of the Hitchin system (2011, background): “From topological mirror symmetry to the fundamental lemma Ngô’s celebrated(3) proof [78] of the fundamental lemma is the culmination of a series of geometrical advances in the understanding of orbital integrals including [59, 36, 65, 66, 67].”
• k-Schur Functions and Affine Schubert Calculus (2013, background): “The necessary modifications were obtained in [58] and [131] respectively.”
• Le lemme fondamental pour les algebres de Lie (2008, background): “[26] Goresky M., Kottwitz R., MacPherson R. : Codimension of root valuation strata. preprint 2006.”
• The fundamental lemma of Jacquet and Rallis (2011, background): “The geometry involved in the Langlands-Shelstad fundamental lemma consists of a local part—the affine Springer fibers (see [5], [6])—and a global part—the Hitchin fibration (see Ngô [11], [12]).”