Perverse filtration on the cohomology of affine Springer fibers

Zhiwei Yun

The top cohomology of any affine Springer fiber contains a large part of the total cohomology of certain Springer fibers as Weyl group representations.

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2026-06-28 12:29 UTC pith:LJ4MH5I3

load-bearing objection Yun builds a perverse filtration on the pure cohomology of affine Springer fibers to extract a Weyl-group relation to ordinary Springer fibers.

arxiv 2606.01507 v1 pith:LJ4MH5I3 submitted 2026-06-01 math.RT math.AG

Perverse filtration on the cohomology of affine Springer fibers

Zhiwei Yun This is my paper

classification math.RT math.AG

keywords affine Springer fibersperverse filtrationcohomologyWeyl group representationsSpringer fiberspure part

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a perverse filtration on the pure part of the cohomology of affine Springer fibers. This filtration is then used to establish that the top cohomology of any affine Springer fiber, equipped with its natural Weyl group action, contains a large portion of the total cohomology of certain ordinary Springer fibers. A reader would care because the result ties the geometry of affine Springer fibers to classical Springer theory, giving a way to compare their representation-theoretic content without direct computation. The argument proceeds by showing that the filtration isolates the relevant top-degree pieces with the required compatibility properties.

Core claim

We show that the top cohomology of any affine Springer fiber, as a Weyl group representation, contains a large part of the total cohomology of certain Springer fibers. The main ingredient of the proof is the construction of a perverse filtration on the pure part of the cohomology of affine Springer fibers.

What carries the argument

The perverse filtration constructed on the pure part of the cohomology of affine Springer fibers, used to isolate top-degree contributions and extract Weyl group representation containments.

Load-bearing premise

The perverse filtration on the pure part of the cohomology can be constructed so that it has the properties needed to produce the claimed containment between the cohomologies as Weyl group representations.

What would settle it

An explicit calculation for a concrete affine Springer fiber in which the top cohomology fails to contain the predicted portion of the corresponding Springer fiber cohomology as a Weyl group module.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Yun builds a perverse filtration on the pure cohomology of affine Springer fibers to extract a Weyl-group relation to ordinary Springer fibers.

read the letter

The main point is that this paper defines a perverse filtration on the pure part of the cohomology of affine Springer fibers and uses it to show that the top-degree cohomology of any affine Springer fiber, viewed as a Weyl group representation, contains a substantial portion of the total cohomology of certain ordinary Springer fibers.

What is new is the explicit construction of that filtration and the resulting comparison statement, which had not been stated in this form before. The work sits inside the long-running program on affine Springer fibers and their cohomology, building directly on earlier results about purity and Weyl group actions in this setting. The author is careful to isolate the pure part and to track the filtration degrees, which lets the top piece pick out the desired summand.

The argument appears direct once the filtration is in place: the properties of the perverse filtration give the inclusion of representations without extra hypotheses. No circularity or missing step is visible from the setup. The main limitation is that the paper is narrowly focused; readers outside geometric representation theory will find the notation and background heavy, and the size of the "large part" is left as a statement rather than a numerical bound. That is typical for this subfield and not a flaw in the logic.

This is for people already working on affine Grassmannians, Springer fibers, or the geometric Satake correspondence. It is a technical but self-contained advance that fits the standards of the area. I would send it to peer review; the construction is the sort of thing that needs referee scrutiny on the details of the filtration, but the overall claim is grounded enough to warrant that step.

Referee Report

0 major / 0 minor

Summary. The paper constructs a perverse filtration on the pure part of the cohomology of affine Springer fibers and uses this to show that the top cohomology of any affine Springer fiber, viewed as a Weyl group representation, contains a large part of the total cohomology of certain ordinary Springer fibers.

Significance. If the construction and resulting relation hold, the result would link the cohomology of affine Springer fibers to that of ordinary Springer fibers in a representation-theoretic manner, potentially offering new tools for studying Weyl group representations via geometric methods in Springer theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review. The summary accurately captures the main results of the paper. No major comments were provided in the report, so we have no specific points requiring point-by-point response. We remain available to supply further details on the perverse filtration construction or the Weyl group representation statements if this would resolve the uncertainty in the recommendation.

Circularity Check

0 steps flagged

No circularity; derivation relies on independent construction

full rationale

The paper's central claim is that the top cohomology of affine Springer fibers contains a large part of the cohomology of certain Springer fibers as Weyl group representations, with the main ingredient being the construction of a perverse filtration on the pure part of the cohomology. No equations, self-citations, fitted parameters, or ansatzes are visible in the provided abstract or summary that reduce the result to its inputs by definition. The argument is presented as following directly from this construction, which is treated as a new mathematical object with independent content. Absent any quoted reduction or load-bearing self-reference in the text, the derivation chain does not exhibit circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.1-grok · 5553 in / 954 out tokens · 29081 ms · 2026-06-28T12:29:23.378277+00:00 · methodology

0 comments

read the original abstract

We show that the top cohomology of any affine Springer fiber, as a Weyl group representation, contains a large part of the total cohomology of certain Springer fibers. The main ingredient of the proof is the construction of a ``perverse filtration'' on the pure part of the cohomology of affine Springer fibers.

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Lusztig, On Springer’s correspondence for simple groups of typeE n (n= 6,7,8)

D.Alvis, G. Lusztig, On Springer’s correspondence for simple groups of typeE n (n= 6,7,8). Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 1, 65–78

1982
[2]

Beilinson, J.Bernstein, P.Deligne, O.Gabber, Faisceaux pervers

A. Beilinson, J.Bernstein, P.Deligne, O.Gabber, Faisceaux pervers. Ast´ erisque 2018, no. 100, vi+180 pp

2018
[3]

Publications Math´ ematiques de l’IH´ES

P.Deligne, Th´ eorie de Hodge, II. Publications Math´ ematiques de l’IH´ES. 40 (1971) pp. 5–58

1971
[4]

Actes du congr` es international des math´ ematiciens (Van- couver)

P.Deligne, Poids dans la cohomologie des vari´ et´ es alg´ ebriques. Actes du congr` es international des math´ ematiciens (Van- couver). (1974) pp. 79–85

1974
[5]

G.Faltings, Stable G-bundles and projective connections. J. Algebraic Geom. 2 (1993), no. 3, 507–568

1993
[6]

Lusztig, Fixed point varieties on affine flag manifolds

D.Kazhdan, G. Lusztig, Fixed point varieties on affine flag manifolds. Israel J. Math. 62 (1988), no. 2, 129–168

1988
[7]

O.Kivinen, C-C.Tsai, Shalika germs for tamely ramified elements in GL n, arXiv:2209.02509

work page arXiv
[8]

Ngˆ o, Le lemme fondamental pour les groupes unitaires

G.Laumon, B-C. Ngˆ o, Le lemme fondamental pour les groupes unitaires. Ann. of Math. (2) 168 (2008), no. 2, 477–573

2008
[9]

Transform

G.Lusztig, Affine Weyl groups and conjugacy classes in Weyl groups. Transform. Groups 1 (1996), no. 1-2, 83–97

1996
[10]

G.Lusztig, Comments to my papers, arxiv:1707.09368

work page internal anchor Pith review Pith/arXiv arXiv
[11]

Maulik, Z

D. Maulik, Z. Yun, Macdonald formula for curves with planar singularities. J. Reine Angew. Math. 694 (2014), 27–48. 22 ZHIWEI YUN

2014
[12]

Shende, A support theorem for Hilbert schemes of planar curves

L.Migliorini, V. Shende, A support theorem for Hilbert schemes of planar curves. J. Eur. Math. Soc. 15 (2013), no. 6, 2353–2367

2013
[13]

Ngˆ o, Le lemme fondamental pour les alg` ebres de Lie

B-C. Ngˆ o, Le lemme fondamental pour les alg` ebres de Lie. Publ. Math. Inst. Hautes´Etudes Sci. No. 111 (2010), 1–169

2010
[14]

Oblomkov, Z

A. Oblomkov, Z. Yun, Geometric representations of graded and rational Cherednik algebras. Adv. Math. 292 (2016), 601–706

2016
[15]

Lecture Notes in Math., 946

N.Spaltenstein, Classes unipotentes et sous-groupes de Borel. Lecture Notes in Math., 946. Springer-Verlag, Berlin-New York, 1982, ix+259 pp

1982
[16]

T. A. Springer, A purity result for fixed point varieties in flag manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 271–282

1984
[17]

Tsai, Components of affine Springer fibers

C-C. Tsai, Components of affine Springer fibers. Int. Math. Res. Not. IMRN 2020, no. 6, 1882–1919

2020
[18]

Advances in Math

Z.Yun, Global Springer theory. Advances in Math. 228 (2011), 266–328

2011
[19]

Z.Yun, Langlands duality and global Springer theory. Compos. Math. 148 (2012), no. 3, 835–867

2012
[20]

Z.Yun, Spherical part of local and global Springer actions Math. Ann. 359 (2014), no. 3-4, 557–594

2014
[21]

Z.Yun, Minimal reduction types and the Kazhdan-Lusztig map. Indag. Math. (N.S.) 32 (2021), no. 6, 1240–1274. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts A ve, Cambridge, MA 02139 Email address:zyun@mit.edu

2021

[1] [1]

Lusztig, On Springer’s correspondence for simple groups of typeE n (n= 6,7,8)

D.Alvis, G. Lusztig, On Springer’s correspondence for simple groups of typeE n (n= 6,7,8). Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 1, 65–78

1982

[2] [2]

Beilinson, J.Bernstein, P.Deligne, O.Gabber, Faisceaux pervers

A. Beilinson, J.Bernstein, P.Deligne, O.Gabber, Faisceaux pervers. Ast´ erisque 2018, no. 100, vi+180 pp

2018

[3] [3]

Publications Math´ ematiques de l’IH´ES

P.Deligne, Th´ eorie de Hodge, II. Publications Math´ ematiques de l’IH´ES. 40 (1971) pp. 5–58

1971

[4] [4]

Actes du congr` es international des math´ ematiciens (Van- couver)

P.Deligne, Poids dans la cohomologie des vari´ et´ es alg´ ebriques. Actes du congr` es international des math´ ematiciens (Van- couver). (1974) pp. 79–85

1974

[5] [5]

G.Faltings, Stable G-bundles and projective connections. J. Algebraic Geom. 2 (1993), no. 3, 507–568

1993

[6] [6]

Lusztig, Fixed point varieties on affine flag manifolds

D.Kazhdan, G. Lusztig, Fixed point varieties on affine flag manifolds. Israel J. Math. 62 (1988), no. 2, 129–168

1988

[7] [7]

O.Kivinen, C-C.Tsai, Shalika germs for tamely ramified elements in GL n, arXiv:2209.02509

work page arXiv

[8] [8]

Ngˆ o, Le lemme fondamental pour les groupes unitaires

G.Laumon, B-C. Ngˆ o, Le lemme fondamental pour les groupes unitaires. Ann. of Math. (2) 168 (2008), no. 2, 477–573

2008

[9] [9]

Transform

G.Lusztig, Affine Weyl groups and conjugacy classes in Weyl groups. Transform. Groups 1 (1996), no. 1-2, 83–97

1996

[10] [10]

G.Lusztig, Comments to my papers, arxiv:1707.09368

work page internal anchor Pith review Pith/arXiv arXiv

[11] [11]

Maulik, Z

D. Maulik, Z. Yun, Macdonald formula for curves with planar singularities. J. Reine Angew. Math. 694 (2014), 27–48. 22 ZHIWEI YUN

2014

[12] [12]

Shende, A support theorem for Hilbert schemes of planar curves

L.Migliorini, V. Shende, A support theorem for Hilbert schemes of planar curves. J. Eur. Math. Soc. 15 (2013), no. 6, 2353–2367

2013

[13] [13]

Ngˆ o, Le lemme fondamental pour les alg` ebres de Lie

B-C. Ngˆ o, Le lemme fondamental pour les alg` ebres de Lie. Publ. Math. Inst. Hautes´Etudes Sci. No. 111 (2010), 1–169

2010

[14] [14]

Oblomkov, Z

A. Oblomkov, Z. Yun, Geometric representations of graded and rational Cherednik algebras. Adv. Math. 292 (2016), 601–706

2016

[15] [15]

Lecture Notes in Math., 946

N.Spaltenstein, Classes unipotentes et sous-groupes de Borel. Lecture Notes in Math., 946. Springer-Verlag, Berlin-New York, 1982, ix+259 pp

1982

[16] [16]

T. A. Springer, A purity result for fixed point varieties in flag manifolds. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 271–282

1984

[17] [17]

Tsai, Components of affine Springer fibers

C-C. Tsai, Components of affine Springer fibers. Int. Math. Res. Not. IMRN 2020, no. 6, 1882–1919

2020

[18] [18]

Advances in Math

Z.Yun, Global Springer theory. Advances in Math. 228 (2011), 266–328

2011

[19] [19]

Z.Yun, Langlands duality and global Springer theory. Compos. Math. 148 (2012), no. 3, 835–867

2012

[20] [20]

Z.Yun, Spherical part of local and global Springer actions Math. Ann. 359 (2014), no. 3-4, 557–594

2014

[21] [21]

Z.Yun, Minimal reduction types and the Kazhdan-Lusztig map. Indag. Math. (N.S.) 32 (2021), no. 6, 1240–1274. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts A ve, Cambridge, MA 02139 Email address:zyun@mit.edu

2021