On the fundamental domain of affine Springer fibers

2013/03/19 by Zongbin Chen, Chen, Zongbin
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1303.4630

openalex publication_date 2013/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For G a connected reductive group, γ∈ \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 \gl₃. In particular, we pave them in affine spaces and we prove the rationality conjecture.

Related

• The geometry of some generalized affine Springer fibers
• Witt vector affine Springer fibers
• Truncated affine Springer fibers and Arthur's weighted orbital integrals
• Components of affine Springer fibers
• Sur le lemme fondamental pour les groupes unitaires