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TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

Part of: Discontinuous groups and automorphic forms Special varieties Lie groups

Published online by Cambridge University Press:  10 November 2021

Zongbin Chen Open the ORCID record for Zongbin Chen [Opens in a new window]
Zongbin Chen*
Affiliation:
Yau Mathematical Science Center, Tsinghua University, Jinchunyuan West Building, Office 255, 100084, Haidian District, Beijing, China
*
zbchen@tsinghua.edu.cn
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Abstract

We explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

Keywords

Weighted orbital integralsAffine Springer fibersHarder-Narasimhan reductionArthur-Kottwitz reduction

MSC classification

Primary: 22E67: Loop groups and related constructions, group-theoretic treatment 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 4 , July 2023 , pp. 1757 - 1818
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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