2023/12/17 by Roman Bezrukavnikov, Bezrukavnikov, Roman, Ivan Karpov +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2312.10582
openalex publication_date 2023/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A key tool for the study of an affine Hecke algebra H is provided by Springer theory of the Langlands dual group via the realization of H as equivariant K-theory of the Steinberg variety. We prove a similar geometric description for Lusztig's asymptotic affine Hecke algebra J identifying it with the sum of equivariant K-groups of the squares of \mathbb C^*-fixed points in the Springer fibers, as conjectured by Qiu and Xi (the same result was also obtained by Oron Popp using different methods). As an application, we give a new geometric proof of Lusztig's parametrization of irreducible representations of J. We also reprove Braverman-Kazhdan's spectral description of J. As another application, we prove a description of the cocenters of H and J conjectured by the first author with Braverman, Kazhdan and Varshavsky. The proof is based on a new algebraic description of J, which may be of independent interest.