Tables and computational data

Graded Kazhdan–Lusztig Atlas

Dan Ciubotaru & Emile Okada

This atlas contains tables of Kazhdan–Lusztig polynomials, Fourier transforms, and microlocal Arthur–ABV packets. The tables record graded nilpotent orbits and their equivariant local systems, orbit closures, and packet membership. They accompany our work on exceptional groups; computations in classical types are also included.

The polynomial calculations use Lusztig’s algorithm for graded Lie algebras. The packet calculations use microlocal vanishing cycles of equivariant perverse sheaves. See About the atlas for the mathematical background, conventions, and citation guidance.

Computed root systems

33 root systems · 1,034 neutral gradings. Expand a family to view the tables.

“Arthur–ABV packets” refers to the microlocal construction. Comparisons with endoscopically defined Arthur packets are discussed separately.

Type A
TypeNeutral gradingsIC parameters across gradingsAvailable results
A₁24KL, Fourier, Arthur–ABV packets
A₂39KL, Fourier, Arthur–ABV packets
A₃525KL, Fourier, Arthur–ABV packets
A₄747KL, Fourier, Arthur–ABV packets
A₅11133KL, Fourier, Arthur–ABV packets
A₆15262KL, Fourier, Arthur–ABV packets
A₇22692KL, Fourier, Arthur–ABV packets
A₈301492KL, Fourier, Arthur–ABV packets
Type B
TypeNeutral gradingsIC parameters across gradingsAvailable results
B₂415KL, Fourier, Arthur–ABV packets
B₃738KL, Fourier, Arthur–ABV packets
B₄13127KL, Fourier, Arthur–ABV packets
B₅21340KL, Fourier, Arthur–ABV packets
B₆351010KL, Fourier, Arthur–ABV packets
B₇552713KL, Fourier, Arthur–ABV packets
B₈867531KL, Fourier, Arthur–ABV packets
B₉13019763KL, Fourier, Arthur–ABV packets
Type C
TypeNeutral gradingsIC parameters across gradingsAvailable results
C₂415KL, Fourier, Arthur–ABV packets
C₃852KL, Fourier, Arthur–ABV packets
C₄14170KL, Fourier, Arthur–ABV packets
C₅24518KL, Fourier, Arthur–ABV packets
C₆401541KL, Fourier, Arthur–ABV packets
C₇644393KL, Fourier, Arthur–ABV packets
C₈10012236KL, Fourier, Arthur–ABV packets
Type D
TypeNeutral gradingsIC parameters across gradingsAvailable results
D₄12122KL, Fourier, Arthur–ABV packets
D₅16245KL, Fourier, Arthur–ABV packets
D₆31863KL, Fourier, Arthur–ABV packets
D₇431982KL, Fourier, Arthur–ABV packets
D₈756001KL, Fourier, Arthur–ABV packets
Type E
TypeNeutral gradingsIC parameters across gradingsAvailable results
E₆21509KL, Fourier, Arthur–ABV packets
E₇451863KL, Fourier, Arthur–ABV packets
E₈704894KL, Fourier, Arthur–ABV packets
Type F
TypeNeutral gradingsIC parameters across gradingsAvailable results
F₄16146KL, Fourier, Arthur–ABV packets
Type G
TypeNeutral gradingsIC parameters across gradingsAvailable results
G₂515KL, Fourier, Arthur–ABV packets