Reducibility of the intersections of components of a Springer fiber

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 ${\textrm {GL}}_{3}$ .

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Singular Localization and Intertwining Functors for Reductive Lie Algebras in Prime Characteristic

Nagoya Mathematical Journal ◽

10.1017/s0027763000009302 ◽

2006 ◽

Vol 183 ◽

pp. 1-55 ◽

Cited By ~ 18

Author(s):

Roman Bezrukavnikov ◽

Ivan Mirković ◽

Dmitriy Rumynin

Keyword(s):

Lie Algebra ◽

Differential Operators ◽

Derived Categories ◽

Central Character ◽

Basic Step ◽

Finite Characteristic ◽

Coherent Sheaves ◽

Springer Fiber ◽

Category Of Modules ◽

Central Characters

In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character λ as sheaves on the partial flag variety corresponding to the singularity of λ. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

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Components of the Springer fiber and domino tableaux

Journal of Algebra ◽

10.1016/s0021-8693(03)00434-4 ◽

2004 ◽

Vol 272 (2) ◽

pp. 711-729 ◽

Cited By ~ 6

Author(s):

Thomas Pietraho

Keyword(s):

Domino Tableaux ◽

Springer Fiber

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On intersections of orbital varieties and components of Springer fiber

Journal of Algebra ◽

10.1016/j.jalgebra.2006.01.029 ◽

2006 ◽

Vol 298 (1) ◽

pp. 1-14 ◽

Cited By ~ 6

Author(s):

A. Melnikov ◽

N.G.J. Pagnon

Keyword(s):

Springer Fiber

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Components of Affine Springer Fibers

International Mathematics Research Notices ◽

10.1093/imrn/rny085 ◽

2018 ◽

Vol 2020 (6) ◽

pp. 1882-1919

Author(s):

Cheng-Chiang Tsai

Keyword(s):

Weyl Group ◽

Characteristic Zero ◽

Reductive Group ◽

Semisimple Element ◽

Orbital Integrals ◽

Number Of Components ◽

Springer Fiber ◽

Springer Fibers

Abstract Let G be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, $\gamma _0\in (\operatorname{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t\gamma _0$. Using methods from p-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.

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