Graded Kazhdan–Lusztig Atlas

About the atlas

Dan Ciubotaru & Emile Okada

Mathematical background

This atlas accompanies our work on Kazhdan–Lusztig polynomials for graded Hecke algebras and Arthur–ABV packets for exceptional groups. The tables describe orbits, equivariant local systems, intersection-cohomology polynomials, Fourier transforms, and packet membership. The underlying geometry is the action of the centralizer of a grading element on the degree-two part of a complex reductive Lie algebra.

We compute the polynomials using Lusztig’s work on perverse sheaves on graded Lie algebras [1], [2], in the explicit formulation of Ciubotaru [3]. Our work extends these computations to exceptional types and their generalized Springer series.

The packet calculations use the microlocal approach of Adams–Barbasch–Vogan [4] and Vogan [5], developed for p-adic groups by Cunningham–Fiori–Moussaoui–Mracek–Xu [6]. Membership in a packet is detected by nonzero microlocal vanishing cycles, or equivalently by the occurrence of the corresponding conormal component in the characteristic cycle.

Coverage and conventions

The index lists types A₁–A₈, B₂–B₉, C₂–C₈, D₄–D₈, G₂, F₄, E₆, E₇, and E₈, with links to the polynomial, Fourier-transform, and packet tables for each listed neutral grading.

For exceptional types, the atlas covers the Dynkin gradings of all nilpotent orbits, including the zero orbit, in both principal and non-principal generalized Springer series.

A table is specified by the complex group and its isogeny type, the weighted Dynkin diagram of the grading, and the generalized Springer series. To read its entries, use the stated conventions for orbit and local-system labels, matrix rows and columns, the polynomial variable, Fourier duality, and packet membership.

Computational assistance

We used ChatGPT and Codex to help develop code, test mathematical ideas, and prepare LaTeX source and documentation. We are responsible for the mathematics, computations, and exposition.

Accompanying papers

The methods and results will be presented in two joint papers, listed here under their working titles:

  1. Dan Ciubotaru and Emile Okada, Kazhdan–Lusztig polynomials for exceptional graded Hecke algebras, in preparation.
  2. Dan Ciubotaru and Emile Okada, Microlocal Arthur packets as a quasi-canonical basis of stable distributions, 2026.

Links and publication details will be added when the manuscripts are available.

Data release and verification

A versioned archive of the tables and computations is forthcoming on Zenodo. The DOI, release date, code links, and reproduction instructions will be added here with the release.

Planned archive contents and checks

The archive is planned to include machine-readable tables, a manifest identifying each dataset by group, grading, and generalized Springer series, and checksums. Reproduction instructions will identify the code version used.

A verification report will record the checks performed on each dataset. Where applicable, these will include diagonal normalization, compatibility with orbit closures, degree bounds, the algorithm’s defining matrix identities, Fourier involutivity, and agreement between characteristic cycles and packet membership. It will also list comparisons with published tables and independent recomputations.

The release will include a version history and contact details for corrections. When reporting an error, please give the version, group, grading, series, and parameter labels.

How to cite

For tables and computations, the preferred citation is the Zenodo release of Dan Ciubotaru and Emile Okada, Graded Kazhdan–Lusztig Atlas. The DOI and BibTeX entry are forthcoming. Until then, please cite the atlas by its authors and title, with the website address and date accessed.

For the mathematical methods and results, please also cite the relevant accompanying paper.

When referring to a specific calculation, give the group, grading (by weighted Dynkin diagram or nilpotent-orbit label), generalized Springer series, and parameter labels. For a polynomial, give its row and column labels in the table.

References

  1. G. Lusztig, Study of perverse sheaves arising from graded Lie algebras, Advances in Mathematics 112 (1995), 147–217.
  2. G. Lusztig, Graded Lie algebras and intersection cohomology. arXiv:math/0604535
  3. D. Ciubotaru, Multiplicity matrices for the affine graded Hecke algebra, Journal of Algebra 320 (2008), 3950–3983. arXiv:0801.1526
  4. J. Adams, D. Barbasch, and D. A. Vogan, Jr., The Langlands Classification and Irreducible Characters for Real Reductive Groups, Birkhäuser, 1992.
  5. D. A. Vogan, Jr., The local Langlands conjecture, in Representation Theory of Groups and Algebras, Contemporary Mathematics 145 (1993), 305–379.
  6. C. Cunningham, A. Fiori, A. Moussaoui, J. Mracek, and B. Xu, Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples, Memoirs of the American Mathematical Society, 2022. arXiv:1705.01885