Pureté des fibres de Springer affines pour $GL_4$

Chen, Zongbin

doi:10.24033/bsmf.2663

Cited by 3 publications

(6 citation statements)

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“…Let γ ∈ t(O) be a regular element. As we show in the appendix of [9], up to conjugation by the Weyl group we can suppose that val(α 12 (γ)) ≤ val(α 23 (γ)), val(α 13 (γ)) = val(α 12 (γ)).…”

Section: Calculations For Gl 3 -Split Casementioning

confidence: 99%

“…Moreover, it is invariant under the right K -action, so it induces a map from to , still denoted by . For , let be the composition As shown in [7, Lemma 6.1], the map is constant on each connected component of , so it has a factorization . A simple calculation of the restriction of the map to shows that the map is just the one induced from the natural inclusion .…”

Section: (Weighted) Orbital Integrals and The Affine Springer Fibersmentioning

confidence: 99%

“…Following Chaudouard and Laumon [7], we define the truncated affine Grassmannian to be We want to point out that its connected components are also parametrized by , but they are not isomorphic in general. However, there is periodicity in the connected components: let be the adjoint group of G and let be the projection induced by the natural projection .…”

Section: Counting Points By Arthur–kottwitz Reductionmentioning

confidence: 99%

“…We reproduce it here for the convenience of the reader. Following Chaudouard and Laumon [7], under the purity assumption the cohomology of can be expressed in terms of its -skeleton under the T -action. Indeed, the T -equivariant cohomology 1 will then be a free -algebra, and we have The torus T acts on with finitely many fixed points, but the -dimensional T -orbits form a higher-dimensional variety which we denote by .…”

Section: Introductionmentioning

confidence: 99%

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Truncated Affine Springer Fibers and Arthur’s Weighted Orbital Integrals

Chen

2021

J. Inst. Math. Jussieu

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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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Section: Calculations For Gl 3 -Split Casementioning

confidence: 99%

Section: (Weighted) Orbital Integrals and The Affine Springer Fibersmentioning

confidence: 99%

Section: Counting Points By Arthur–kottwitz Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Truncated Affine Springer Fibers and Arthur’s Weighted Orbital Integrals

Chen

2021

J. Inst. Math. Jussieu

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“…For the proof, the reader can refer to [5], corollaire 2.3. For μ ∈ X T n , let C(μ) = X n ∩ B μ K /K .…”

Section: Of the Poincaré Series Lemma 54 Let ξ ∈ A Gmentioning

confidence: 99%

The $$\xi $$ ξ -stability on the affine grassmannian

Chen

2015

Math. Z.

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We introduce a notion of ξ-stability on the affine grassmannian X for the classical groups, this is the local version of the ξ-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient X ξ /T of the stable part X ξ by the maximal torus T exists as an ind-k-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles over an algebraic curve. For the group SL d , we calculate the Poincaré series of the quotient X ξ /T .

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Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

Turner

2024

International Mathematics Research Notices

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We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

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Pureté des fibres de Springer affines pour $GL_4$

Cited by 3 publications

References 0 publications

Truncated Affine Springer Fibers and Arthur’s Weighted Orbital Integrals

Truncated Affine Springer Fibers and Arthur’s Weighted Orbital Integrals

The $$\xi $$ ξ -stability on the affine grassmannian

Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

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