Cyclotomic level maps and associated varieties of simple affine vertex algebras

Peng Shan; Qixian Zhao; WenBin Yan

REVIEW 3 major objections 5 minor 4 cited by

Cyclotomic level maps and associated varieties of simple affine vertex algebras

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A cyclotomic level map on nilpotent orbits, compatible with a map on Weyl group classes, conjecturally describes the associated variety of every simple affine vertex algebra at integer levels above criticality.

desk verdict The paper proves a clean compatibility theorem and a cell bijection, then attaches a promising but unproven sheet conjecture to a foundation that is only atlas-checked in exceptional types. read the letter →

arxiv 2507.09254 v2 pith:XLQGBMFS submitted 2025-07-12 math.RT math-phmath.MPmath.QA

Peng Shan , Wenbin Yan , Qixian Zhao This is my paper · ORCID

classification math.RTmath-phmath.MPmath.QA MSC 17B6917B0817B67

keywords affinevertexalgebrasassociatedvarietiesnilpotentorbitscyclotomiclevelmapsWeylgroupconjugacyclassestwo-sidedcellsquasi-lissesheets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a single integer attached to each nilpotent orbit—the cyclotomic level cl_n—and a matching integer cl_W attached to each conjugacy class in the Weyl group, and proves that the two agree under the minimal reduction type map (equivalently, Lusztig's map). Using these numbers, the paper conjectures that for every simply-laced Lie algebra and every integer level k=m-h^vee above criticality, the associated variety of the simple affine vertex algebra L_k(g) is an explicitly constructed sheet S(l,dO_L) determined by the orbit O(m) whose cyclotomic level is at most m. If the conjecture is right, an invariant that was previously understood only in scattered cases becomes uniform and algorithmic, and quasi-lisseness of L_k is read off from whether O(m) is distinguished. All associated varieties that had been computed before are reproduced by the conjecture.

What carries the argument

The load-bearing object is the cyclotomic level map cl_n on nilpotent orbits, together with its Weyl-group analogue cl_W. For an orbit, cl_n(e)=a+1 with 2a the largest h-weight on the Bala-Carter Levi; equivalently cl_n(e) is the least m with (ad e)^{2m}=0. For a conjugacy class, cl_W([w]) is the largest m whose cyclotomic polynomial divides the characteristic polynomial of w on the reflection representation. The map cl_n attaches to each positive integer m the unique maximal orbit O(m) in {cl_n≤m}; the paper's formula for O(m) is the identity O(m)=∪_{cl_n(O)≤m} O. The compatibility theorem is proved by relating both maps to minimal reduction types through root valuation strata, and the conjecture packages the output as a sheet S(l,dO_L)=closure of the image of G×^P(dO_L×z(l)×u), where P=LU is a parabolic containing the Bala-Carter Levi L.

What would settle it

For g=E6 at k=-5 (so m=7), compute the associated variety of L_k(E6) directly from Zhu's C2-algebra and compare it with the sheet S(l,dO_L) that Conjecture 2.4.1 attaches to O(7)=E6(a3). This level is not among the previously computed cases in Table 2, so any mismatch would be a decisive counterexample.

Watch

Extended reading notes

Core claim

The paper's central claim is that two a priori unrelated combinatorial invariants coincide and together organize the associated varieties of simple affine vertex algebras. On nilpotent orbits, cl_n(e) is defined as a+1 where 2a is the largest h-weight of an sl2-triple in the Bala-Carter Levi, equivalently cl_n(e)=min{m | (ad e)^{2m}=0}; on Weyl group classes, cl_W([w]) is the largest order of an eigenvalue of w on the reflection representation, equivalently the largest m with Φ_m dividing the characteristic polynomial. Theorem 3.3.5 proves cl_n(RTmin([w]))=cl_W([w]) for the minimal reduction type map, and Theorem 4.2.1 proves that Lusztig's bijection sends the two-sided cell c(w_m), attached to the dominant affine translate of mΛ0+ρ, to the orbit O(m). Conjecture 2.4.1 then claims that for simply-laced g and k=m-h^vee with m≥1, the associated variety of L_k(g) equals S(l,dO_L), the sheet attached to the Barbasch-Vogan dual of the distinguished factor of O(m).

Load-bearing premise

The construction depends on the type-by-type lemma that the largest h-eigenvalue on the whole Lie algebra exceeds that on a Bala-Carter Levi by at most one; in the exceptional types that lemma is verified by computer, not by a written proof.

Editorial extensions

If this is right

Every integer level k=m-h^vee above criticality for simply-laced g gets a predicted associated variety, replacing the previous case-by-case computations.
The quasi-lisse property of L_k(g) becomes a combinatorial condition: it holds exactly when O(m) is distinguished.
The associated varieties of the quantized Drinfeld-Sokolov reductions H^0_f(L_k(g)) are determined by intersecting the predicted sheet with Slodowy slices.
The two-sided cell c(w_m) in the affine Weyl group corresponds under Lusztig's bijection to O(m), so the same integer m organizes cells, orbits, and vertex algebra varieties.
Lusztig's bijection restricted to orbits O(m) can be computed by the explicit recipe O(m) -> ξ_m -> c(w_m), mirroring the classical Barbasch-Vogan construction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the conjecture holds, the same integer m governs three a priori separate objects: the sheet X_{L_k}, the minimal reduction type of the class [w], and the two-sided cell c(w_m); that three-way correspondence is a natural place to look for a Langlands-style formulation.
The partition descriptions of O(m) in classical types suggest that the associated variety of L_k in types A through D can be written purely combinatorially, which would make the conjecture checkable by partition algorithms without orbit tables.
The Kac diagrams tabulated for ξ_m in the exceptional types are the same kind of data used in periodic gradings and epipelagic representations; exploiting that connection could give a uniform proof of the key h-weight lemma in exceptional types rather than a computer check.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real news here is the compatibility theorem (3.3.5) and the two-sided cell statement (4.2.1). The cyclotomic level map cl_n on nilpotent orbits is not new — the paper says so, citing KL88 and Ree+12 — but proving that cl_W on Weyl classes matches cl_n through the minimal reduction type map is a solid, original result. The classical-type proofs are explicit and checkable, and the Kac diagram tables for xi_m are useful on their own. The conjecture on associated varieties is clearly labeled as a conjecture, checked against all known cases, and not oversold. Citation pattern looks honest.

The soft spot is exactly where the reader's stress-test lands: Lemma 2.1.3, which clamps the highest h-weight on g to be at most one more than on the Bala-Carter Levi, is load-bearing. It gives the alternative description of cl_n as the Jordan block height, which then defines the orbits O(m) that feed Conjecture 2.4.1. In E6, E7, E8, F4, and G2 the proof is only "verified using the atlas software," with no transcripts, parameter values, or independent certificates. If one exceptional value were wrong, O(m) would shift and the predicted sheets would move. I don't think the lemma is false — atlas is generally trustworthy — but from the manuscript alone I cannot verify it. That is a genuine reproducibility gap in the foundation, not a convenience issue. The same reliance on atlas appears in the proof of Theorem 4.2.1 for type E.

Proportionately: the proven structural results in classical types seem sound, and the conjecture is plausible and well-motivated. The main weakness is verifiability of exceptional-type data. For a paper whose central conjecture depends on those values, I would want the atlas output shipped or replaced by a human-readable check before publication.

This deserves a serious referee. A representative from vertex algebras can assess the conjecture's traction, and a Lie theorist with atlas access can close the gap. I would bring it to reading group and would cite the compatibility theorem. Verdict: engage, but require supplemental data for the exceptional case.

Referee Report

3 major / 5 minor

Summary. The paper introduces a cyclotomic level map cln on nilpotent orbits and a companion map clW on Weyl group conjugacy classes, proves that they are compatible with Yun's minimal reduction type map, and proves a relation between Lusztig's bijection for two-sided cells and the orbits O(m) attached to the levels. On this basis the authors formulate Conjecture 2.4.1, proposing that for simply-laced g and integer levels k=m-h^vee the associated variety X_{L_k(g)} is the sheet S(l, dO_{L}) determined by the Bala-Carter Levi data of O(m). The classical-type proofs are detailed, and the conjecture is checked against all available computations of associated varieties, including the recent results of Arakawa-Futorny-Krizka.

Significance. If the main conjecture is correct, it would give the first uniform description of associated varieties of simple affine vertex algebras at all non-admissible integer levels above criticality, together with a quasi-lisseness criterion. The paper's unconditional results are also valuable: Theorem 3.3.5 gives a clean compatibility statement between the two new level maps and the minimal reduction type map, and Theorem 4.2.1 connects the combinatorial construction of O(m) with Lusztig's two-sided cells. The classical-type computations in Sections 5.1 and 5.2 are explicit and self-contained. The main weakness is that the exceptional-type cases rest on atlas software output that is neither reproduced nor machine-checked in the manuscript; this is a verifiability gap in a load-bearing part of the construction, not merely a cosmetic omission.

major comments (3)

[Section 5.1, Lemma 5.1.1 (Lemma 2.1.3)] The second inequality in Lemma 2.1.3 is verified only via the atlas software in the exceptional types E6, E7, E8, F4, and G2, but no scripts, transcripts, parameter values, or independent certificates are supplied. This lemma is load-bearing: it yields Lemma 2.1.4 (cln(e) as the minimal m with (ad e)^{2m}=0), enters Proposition 3.3.7 and Theorem 3.3.5, and determines the orbit O(m) used in Conjecture 2.4.1. A single wrong value in Figures 1-5 would shift some O(m) and change the predicted sheet. I ask the authors to ship the atlas input/output or an independent root-system certificate for every exceptional distinguished orbit.
[Section 4.2, proof of Theorem 4.2.1] The type E verification of Theorem 4.2.1 is again delegated to the atlas software ('the calculations in type E are done by utilizing the atlas software'), and Tables 6-8 list the Kac diagrams and parabolic types W_m without derivation. Unlike the detailed type A and D arguments in Section 5.2, the exceptional cases cannot be checked from the paper. Since Theorem 4.2.1 is one of the two main theorems and is used as conceptual evidence for Conjecture 2.4.1, this gap should be closed by providing the relevant atlas scripts or tabulated intermediate data.
[Section 2.1, Theorem 2.1.6 and Figures 1-5] Theorem 2.1.6(4) is quoted from [Geo04] and the partition formulas for O(m) in classical types are quoted without proof; this is acceptable as citation to published work. However, the exceptional-type lists of cln and O(m) in Figures 1-5 are outputs of the same atlas computation that underlies Lemma 2.1.3. The paper should state explicitly which entries are consequences of the written classical-type proof and which entries depend on the unverified software computation, so that the reader can assess the confidence level of the conjecture in exceptional types.

minor comments (5)

[Section 3.3, Theorem 3.3.5] The codomain of cln is written with a corrupted notation that appears as 'Z1...h'; this should be corrected to a standard notation such as the positive integers or the interval from 1 to h.
[Section 2.1.13 and List of Figures] Figures 1-5 are essential data for the exceptional-type values of cln and O(m), but the figure images are not reproduced in the manuscript text. Please ensure the final submission contains all five figures with legible labels.
[Section 1, Corollary 1.0.3] The heading 'Corollary of Conjecture, 1.0.3' should be renumbered as a formal corollary (for example, 'Corollary 1.0.3') rather than left as a run-in phrase.
[Section 5.2.15] In the final verification cases for type D_n, the commutative diagram involving KL is cited before all symbols in the diagram are introduced; consider moving the definitions of the parahoric and Levi subgroups immediately before the displayed diagram.
[References] The entries [SYZa] and [SYZb] are listed as 'in preparation'; this is acceptable, but the bibliography should mark them consistently as unpublished work to avoid confusion with the cited arXiv preprints.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cyclotomic level maps are defined from intrinsic Lie-theoretic data (Bala–Carter Levis, sl2-triples, eigenvalue orders) and the conjecture is a genuine prediction checked against independent computations.

full rationale

The paper's central constructions are not equivalent to their inputs by construction. The cyclotomic level map cln is defined in Definition 2.1.2 purely from a Bala–Carter Levi l, an sl2-triple, and the largest h-weight on l; it does not take associated varieties as input. The alternative description cln(e)=min{m | (ad e)^{2m}=0} in Lemma 2.1.4 is derived from the structural definition together with Lemma 2.1.3, and the remarkable set identity O(m)=union over cln(O)<=m of O is quoted from the independent work [Geo04] (Theorem 2.1.6), not derived from the conjecture. The Weyl-group cyclotomic level map clW in Definition 3.3.1 is likewise defined independently from the maximal order of eigenvalues on the reflection representation. The compatibility clW = cln ∘ RTmin (Theorem 3.3.5) is proved from the root-valuation machinery and Proposition 3.3.7, not assumed in the definition. Conjecture 2.4.1 is a genuine prediction: the associated variety XLk is asserted to equal a sheet S(l,dO_L) built from O(m), and the paper verifies this against previously computed cases from [GK07], [AM17], [AM18a], [AM18b], [AFK24], and [JS25] in Tables 1–3 of Section 2.4. These checks compare the prediction with independent results rather than fitting parameters. The one limitation worth flagging is in the proof of Lemma 2.1.3 in Section 5.1, where the exceptional cases are only 'verified using the atlas software' with no reproduced transcripts or certificates; this is a verifiability gap in the foundation of the construction, as the skeptic notes, but it is not a circular reduction. The atlas output is an external computational check, not a restatement of the associated-variety predictions, and failure of the lemma would change the definition of O(m) rather than reveal that the conjecture was presupposed. The self-citations [SXY24] and the forthcoming [SYZa], [SYZb] are contextual or motivational and not load-bearing for the main theorem. Thus no step in the paper reduces, by the paper's own equations or by self-citation, to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The map cl_n is a computed invariant, not a fit. The conjecture is not derived from data: it is stated and then checked against preexisting computations. The only computational reliance is the atlas verification in exceptional types, which is not shipped as code.

assumptions (5)

standard math Associated varieties of simple affine vertex algebras are closed conic G-invariant Poisson subschemes of g, and are known for admissible levels (Ara15).
Used in Section 2.3 to set up X_{L_k} and to compare with Conjecture 2.4.1; the known cases cited in Tables 2 and 3 form the evidence base.
standard math Yun's theorem: the minimal reduction type map RTmin equals Lusztig's map Phi (Theorem 3.2.6).
This identification is the bridge used in Theorem 3.3.5; the paper cites Yun25 rather than proving it.
standard math Lusztig's bijection between two-sided cells in the affine Weyl group and nilpotent orbits of the dual group.
Theorem 4.2.1 relies on this bijection, recalled in Section 4.1.7.
domain assumption Lemma 2.1.3: for an sl2-triple with highest h-weight on a Bala-Carter Levi l equal to 2a, the highest h-weight on g is 2a or 2a+1.
Proven type-by-type; exceptional types are checked with the atlas software, not by a written proof.
standard math Barbasch-Vogan-Lusztig-Spaltenstein duality d maps distinguished orbits of the dual Lie algebra to special orbits, and the sheet construction S(l,dO_L) is the appropriate closure.
Used throughout Conjecture 2.4.1; the properties are cited from BV85, Lus79, Spa82.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Cyclotomic level maps and associated varieties of simple affine vertex algebras." pith.science (2026). https://pith.science/paper/XLQGBMFS

@misc{pith2026250709254,
  author       = {Pith},
  title        = {Pith review of: Cyclotomic level maps and associated varieties of simple affine vertex algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLQGBMFS}},
  note         = {Machine review of arXiv:2507.09254}
}

abstract

In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{\mathcal{N}}$ in a complex semi-simple Lie algebra $\mathfrak{g}$ and the set of conjugacy classes $\underline{W}$ in its Weyl group, with values in positive integers. We show that these maps are compatible under Lusztig's map $\underline{W} \to \underline{\mathcal{N}}$, which is also the minimal reduction type map as shown by Yun. We also discuss their relationship with two-sided cells in affine Weyl groups. We use these maps to formulate a conjecture on the associated varieties of simple affine vertex algebras attached to $\mathfrak{g}$ at non-admissible integer levels, and provide some evidence for this conjecture.

Figures

Figures reproduced from arXiv: 2507.09254 by the authors.

**Figure 1.** Values of cln in E6 0 [1] A1 2 2A1 2 3A1 [2] A2 3 A2+A1 3 2A2 3 A2+2A1 3 A3 4 2A2+A1 [3] A3+A1 4 D4(a1) [4] A4 5 D4 6 A4+A1 [5] A5 6 D5(a1) 6 E6(a3) [6] D5 [8] E6(a1) [9] E6 [12] [PITH_FULL_IMAGE:figures/full_fig_p042_1.png] view at source ↗

**Figure 2.** Values of cln in E7 0 [1] A1 2 2A1 2 (3A1)” 2 (3A1)’ 2 A2 3 4A1 [2] A2+A1 3 A2+2A1 3 A3 4 2A2 3 A2+3A1 3 (A3+A1)” 4 2A2+A1 [3] (A3+A1)’ 4 D4(a1) 4 A3+2A1 4 D4 6 D4(a1)+A1 4 A3+A2 4 A4 5 A3+A2+A1 [4] (A5)” 6 D4+A1 6 A4+A1 5 D5(a1) 6 A4+A2 [5] (A5)’ 6 A5+A1 6 D5(a1)+A1 6 D6(a2) 6 E6(a3) 6 D5 8 E7(a5) [6] A6 [7] D5+A1 8 D6(a1) 8 E7(a4) [8] D6 10 E6(a1) [9] E6 12 E7(a3) [10] E7(a2) [12] E7(a1) [14] E7 [18] [PITH_FULL_I… view at source ↗

**Figure 3.** Values of cln in E8, part I E8(a7) [6] A6 7 D6(a1) 8 A6+A1 [7] E7(a4) 8 E6(a1) 9 D5+A2 8 D6 10 E6 12 D7(a2) 8 A7 [8] E6(a1)+A1 9 E7(a3) 10 E8(b6) [9] D7(a1) 10 E6+A1 12 E7(a2) 12 E8(a6) [10] D7 12 E8(b5) 12 E7(a1) 14 E8(a5) [12] E8(b4) [14] E7 18 E8(a4) [15] E8(a3) [18] E8(a2) [20] E8(a1) [24] E8 [30] [PITH_FULL_IMAGE:figures/full_fig_p044_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Values of cln in E8, part II 0 [1] A1 2 2A1 2 3A1 2 A2 3 4A1 [2] A2+A1 3 A2+2A1 3 A3 4 A2+3A1 3 2A2 3 2A2+A1 3 A3+A1 4 D4(a1) 4 D4 6 2A2+2A1 [3] A3+2A1 4 D4(a1)+A1 4 A3+A2 4 A4 5 A3+A2+A1 4 D4+A1 6 D4(a1)+A2 4 A4+A1 5 2A3 [4] D5(a1) 6 A4+2A1 5 A4+A2 5 A5 6 D5(a1)+A1 6 …

**Figure 5.** Figure 5: Values of cln in F4 and G2 0 [1] A1 2 A1s 2 A1+A1s [2] A2 3 A2s 3 A2+A1s 3 B2 4 A2s+A1 [3] C3(a1) 4 F4(a3) [4] B3 6 C3 6 F4(a2) [6] F4(a1) [8] F4 [12] 0 [1] A1 2 A1s [2] G2(a1) [3] G2 [6] [PITH_FULL_IMAGE:figures/full_fig_p046_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 20 citations worldwide. Full citation record

Subregular affine cells and the level $-1$ vertex algebra of type $D$
math.QA 2026-08 conditional novelty 7.0 of 10

For every l >= 5, the vacuum block of the level -1 simple affine vertex algebra of type D_l has exactly l+1 irreducible modules, indexed by a subregular Kazhdan-Lusztig left cell.
Completing the Arakawa--Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras
math.QA 2026-07 accept novelty 7.0 of 10

All open cases of the Arakawa–Moreau maximal-ideal conjecture are proved: the prescribed singular vectors generate the maximal ideals at level -1 for D_l and at negative levels n>0 for D4, E6, E7, E8.
Associated varieties of simple affine vertex algebras at rational levels
math.RT 2026-06 unverdicted novelty 7.0 of 10

Authors conjecture associated varieties of L_k(g) for simply-laced g at rational k > critical level, using Gao-Liu-Lo-Shahidi covering duality map, and supply supporting evidence.
Affine vertex algebras and an affine analog of Barbasch-Vogan's construction
math.RT 2026-08 conditional novelty 5.0 of 10

This expository preprint proposes a Langlands-style conjectural description of associated varieties and simple modules of simple affine vertex algebras at integer levels in simply-laced types.

Reference graph

Works this paper leans on

10 extracted references · 5 canonical work pages · cited by 4 Pith papers

[1]

Introduction to W-algebras and their representation theory

[Ara15] T. Arakawa. “Associated varieties of modules over Kac-Moody algebras andC2-cofiniteness of W-algebras”. In:Int. Math. Res. Not. IMRN22 (2015), pp. 11605–11666.doi:10 . 1093 / imrn / rnu277. [Ara17] T. Arakawa.Introduction to W-algebras and their representation theory. 2017.doi:10.48550/arxiv. 1605.00138. arXiv:1605.00138 [math.RT]. [AFK24] T. Arak...

work page Pith review arXiv doi:10.48550/arxiv.1605.00138 2015
[7]

Associated varieties of simple affine VOAsL k(sl3)andW-algebras Wk(sl3,f)

2023.doi: 10.48550/arxiv.2301.10967. arXiv:2301.10967 [math.AG]. [JS25] C. Jiang and J. Song. “Associated varieties of simple affine VOAsL k(sl3)andW-algebras Wk(sl3,f)”. In:Comm. Math. Phys.406.5 (2025), Paper No. 104, 34.doi:10 . 1007 / s00220 - 025-05291-9. [Jos85] A.Joseph.“Ontheassociatedvarietyofaprimitiveideal”.In:J.Algebra93.2(1985),pp.509–523. do...

arXiv doi:10.48550/arxiv.2301.10967 2025
[10]

Quantum reduction for affine superalgebras

University Lecture Series. American Mathematical Society, Providence, RI, 1997, pp. viii+141.doi:10.1090/ulect/010. [KRW03] V. Kac, S.-S. Roan, and M. Wakimoto. “Quantum reduction for affine superalgebras”. In:Comm. Math. Phys.241.2-3 (2003), pp. 307–342.doi:10.1007/s00220-003-0926-1. [KW89] V. G. Kac and M. Wakimoto. “Classification of modular invariant ...

doi:10.1090/ulect/010 2003
[11]

Minimal reduction type and the Kazhdan-Lusztig map

arXiv:2010.13642 [math.RT]. [Zhu96] Y. Zhu. “Modular Invariance of Characters of Vertex Operator Algebras”. In:Journal of the American Mathematical Society9.1 (1996), pp. 237–302. List of Tables 1 Associated varieties ofL k(D4)predicted by Conjecture 2.4.1 9 2 Known cases of Conjecture 2.4.1 10 3 [AFK24] versus Conjecture 2.4.1 11 4 Kac diagrams ofξ m inA...

work page Pith review arXiv 1996
[30]

Unipotentrepresentationsofcomplexsemisimplegroups

[BV85] D.BarbaschandD.A.VoganJr.“Unipotentrepresentationsofcomplexsemisimplegroups”.In: Ann. of Math. (2)121.1 (1985), pp. 41–110.doi:10.2307/1971193. [Bee+15] C.Beemetal.“Infinitechiralsymmetryinfourdimensions”.In:Comm.Math.Phys.336.3(2015), pp. 1359–1433.doi:10.1007/s00220-014-2272-x. [Bez09] R. Bezrukavnikov. “Perverse sheaves on affine flags and nilpo...

work page arXiv 1985
[88]

Vertex operator algebras associated to representations of affine and Virasoro algebras

Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004, pp. xiv+400. doi:10.1090/surv/088. [FZ92] I. B. Frenkel and Y. Zhu. “Vertex operator algebras associated to representations of affine and Virasoro algebras”. In:Duke Mathematical Journal66.1 (1992), pp. 123–168.doi:10.1215/S0012- 7094-92-06604-X. [Geo04] U. of Georgi...

doi:10.1090/surv/088 1992
[177]

OnrationalityofW-algebras

[KW08] V.G.KacandM.Wakimoto.“OnrationalityofW-algebras”.In:TransformationGroups13.3(2008), pp. 671–713. [Kac69] V.G.Kac.“AutomorphismsoffiniteorderofsemisimpleLiealgebras”.In:Funktsional’nyiAnaliz i ego Prilozheniya3.3 (1969), pp. 94–96. [KT84] M. Kashiwara and T. Tanisaki. “The characteristic cycles of holonomic systems on a flag mani- fold”. In:Inventio...

work page doi:10.48550/arxiv.2108.03453 2008
[236]

Translated from the 1995 Japanese edition by Takeuchi

Progress in Mathematics. Translated from the 1995 Japanese edition by Takeuchi. Birkhäuser Boston, Inc., Boston, MA, 2008, pp. xii+407.doi:10.1007/978-0-8176-4523-6. [JY23] K. Jakob and Z. Yun.A Deligne-Simpson problem for irregularG-connections overP

doi:10.1007/978-0-8176-4523-6 1995

Show all 10 references

[326]

Sheets and associated varieties of affine vertex algebras

Progr. Math. Birkhäuser/Springer, Cham, 2018, pp. 41–57.doi:10.1007/978-3-030-02191-7\_2. [AM17] T. Arakawa and A. Moreau. “Sheets and associated varieties of affine vertex algebras”. In:Adv. Math.320 (2017), pp. 157–209.doi:10.1016/j.aim.2017.08.039. [AM18a] T.ArakawaandA.Mor...

2017 doi
[2025]

Quantization of the Drinfeld-Sokolov reduction

arXiv:2504.05113 [math.RT]. [CM93] D.H.CollingwoodandW.M.McGovern.NilpotentorbitsinsemisimpleLiealgebras.VanNostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993, pp. xiv+186. [FF90] B. Feigin and E. Frenkel. “Quantization of the Drinfeld-Sokolov redu...

1990 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Introduction to W-algebras and their representation theory

[Ara15] T. Arakawa. “Associated varieties of modules over Kac-Moody algebras andC2-cofiniteness of W-algebras”. In:Int. Math. Res. Not. IMRN22 (2015), pp. 11605–11666.doi:10 . 1093 / imrn / rnu277. [Ara17] T. Arakawa.Introduction to W-algebras and their representation theory. 2017.doi:10.48550/arxiv. 1605.00138. arXiv:1605.00138 [math.RT]. [AFK24] T. Arak...

work page Pith review arXiv doi:10.48550/arxiv.1605.00138 2015

[2] [7]

Associated varieties of simple affine VOAsL k(sl3)andW-algebras Wk(sl3,f)

2023.doi: 10.48550/arxiv.2301.10967. arXiv:2301.10967 [math.AG]. [JS25] C. Jiang and J. Song. “Associated varieties of simple affine VOAsL k(sl3)andW-algebras Wk(sl3,f)”. In:Comm. Math. Phys.406.5 (2025), Paper No. 104, 34.doi:10 . 1007 / s00220 - 025-05291-9. [Jos85] A.Joseph.“Ontheassociatedvarietyofaprimitiveideal”.In:J.Algebra93.2(1985),pp.509–523. do...

arXiv doi:10.48550/arxiv.2301.10967 2025

[3] [10]

Quantum reduction for affine superalgebras

University Lecture Series. American Mathematical Society, Providence, RI, 1997, pp. viii+141.doi:10.1090/ulect/010. [KRW03] V. Kac, S.-S. Roan, and M. Wakimoto. “Quantum reduction for affine superalgebras”. In:Comm. Math. Phys.241.2-3 (2003), pp. 307–342.doi:10.1007/s00220-003-0926-1. [KW89] V. G. Kac and M. Wakimoto. “Classification of modular invariant ...

doi:10.1090/ulect/010 2003

[4] [11]

Minimal reduction type and the Kazhdan-Lusztig map

arXiv:2010.13642 [math.RT]. [Zhu96] Y. Zhu. “Modular Invariance of Characters of Vertex Operator Algebras”. In:Journal of the American Mathematical Society9.1 (1996), pp. 237–302. List of Tables 1 Associated varieties ofL k(D4)predicted by Conjecture 2.4.1 9 2 Known cases of Conjecture 2.4.1 10 3 [AFK24] versus Conjecture 2.4.1 11 4 Kac diagrams ofξ m inA...

work page Pith review arXiv 1996

[5] [30]

Unipotentrepresentationsofcomplexsemisimplegroups

[BV85] D.BarbaschandD.A.VoganJr.“Unipotentrepresentationsofcomplexsemisimplegroups”.In: Ann. of Math. (2)121.1 (1985), pp. 41–110.doi:10.2307/1971193. [Bee+15] C.Beemetal.“Infinitechiralsymmetryinfourdimensions”.In:Comm.Math.Phys.336.3(2015), pp. 1359–1433.doi:10.1007/s00220-014-2272-x. [Bez09] R. Bezrukavnikov. “Perverse sheaves on affine flags and nilpo...

work page arXiv 1985

[6] [88]

Vertex operator algebras associated to representations of affine and Virasoro algebras

Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004, pp. xiv+400. doi:10.1090/surv/088. [FZ92] I. B. Frenkel and Y. Zhu. “Vertex operator algebras associated to representations of affine and Virasoro algebras”. In:Duke Mathematical Journal66.1 (1992), pp. 123–168.doi:10.1215/S0012- 7094-92-06604-X. [Geo04] U. of Georgi...

doi:10.1090/surv/088 1992

[7] [177]

OnrationalityofW-algebras

[KW08] V.G.KacandM.Wakimoto.“OnrationalityofW-algebras”.In:TransformationGroups13.3(2008), pp. 671–713. [Kac69] V.G.Kac.“AutomorphismsoffiniteorderofsemisimpleLiealgebras”.In:Funktsional’nyiAnaliz i ego Prilozheniya3.3 (1969), pp. 94–96. [KT84] M. Kashiwara and T. Tanisaki. “The characteristic cycles of holonomic systems on a flag mani- fold”. In:Inventio...

work page doi:10.48550/arxiv.2108.03453 2008

[8] [236]

Translated from the 1995 Japanese edition by Takeuchi

Progress in Mathematics. Translated from the 1995 Japanese edition by Takeuchi. Birkhäuser Boston, Inc., Boston, MA, 2008, pp. xii+407.doi:10.1007/978-0-8176-4523-6. [JY23] K. Jakob and Z. Yun.A Deligne-Simpson problem for irregularG-connections overP

doi:10.1007/978-0-8176-4523-6 1995