Atlas of Lie Groups and Representations

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atlas Library · Script File

show_nilpotent_orbits.at

Source
atlas-scripts/show_nilpotent_orbits.at (547 lines)
Definitions
35
Loads
Loaded by
weak_packets.at

Definitions in source order

Commented-out code, lines 4–12 (7 lines)
set real_forms_central_invariant(InnerClass ic,ratvec v)=[int]:
##for G@i in ic.real_forms do if equal_central_invariant(G,v)
 then [i] else [] fi od

set strong_real_forms_central_invariant(InnerClass ic,ratvec v)=[int]:
##for i in ic.strong_real_forms do if equal_central_invariant(ic.real_forms[i],v)
 then [i] else [] fi od

strong_real_forms_nilpotent_orbit_correct_central_invariant

L14strong_real_forms_nilpotent_orbit_correct_central_invariant(ComplexNilpotent O, InnerClass ic) = [RealForm]

strong_real_forms_nilpotent_orbit_e

L18strong_real_forms_nilpotent_orbit_e(ComplexNilpotent O, InnerClass ic) = [RealForm]
Commented-out code, lines 22–26 (3 lines)
set number_real_nilpotent_orbits_correct_central_invariant(ComplexNilpotent O_check,InnerClass ic_check)=[int]:
for G in strong_real_forms_nilpotent_orbit_correct_central_invariant(O_check,ic_check) do
number_real_nilpotent_orbits(O_check,G) od

number_real_nilpotent_orbits_correct_central_invariant

L32number_real_nilpotent_orbits_correct_central_invariant(ComplexNilpotent O_check,InnerClass ic) = [int]
run over real forms with correct central invariant
for each such rf, compute number of real forms of orbit, return [int]

debug

L52debug(ComplexNilpotent O_check,InnerClass ic)

compact_inner_class

L61compact_inner_class(RootDatum rd)
for k in srf_int_correct_type do number_real_nilpotent_orbits(O_check_int,rf_int[k]) od

show_nilpotent_orbits_short

L64show_nilpotent_orbits_short([ComplexNilpotent] orbits) = void
compact listing: i H diagram dim BC-Levi Cent A(O)

also in this file at line 90, line 176, line 370, line 462, line 493

show_nilpotent_orbits 2 overloads

L76show_nilpotent_orbits([ComplexNilpotent] orbits) = void
compact listing: i H diagram dim BC-Levi Cent A(O)
L89show_nilpotent_orbits(RootDatum rd) = void

also in this file at line 177, line 371, line 477, line 492

show_nilpotent_orbits_short

L90show_nilpotent_orbits_short(RootDatum rd) = void

also in this file at line 64, line 176, line 370, line 462, line 493

show_nilpotent_orbits_long 2 overloads

L92show_nilpotent_orbits_long([ComplexNilpotent] orbits) = void
L108show_nilpotent_orbits_long(RootDatum rd) = void

also in this file at line 114, line 144, line 180, line 308, line 338, line 374, line 376, line 495, line 512, line 515

show commands for orbits of RealForm G L110

includes: number of real forms of the orbit
L111

show_nilpotent_orbits_long 2 overloads

L114show_nilpotent_orbits_long([ComplexNilpotent] orbits,RealForm G) = void
compact listing: i H diagram dim BC-Levi Cent A(O) #RF C_2
L144show_nilpotent_orbits_long([ComplexNilpotent] orbits,InnerClass ic) = void

also in this file at line 92, line 108, line 180, line 308, line 338, line 374, line 376, line 495, line 512, line 515

show_nilpotent_orbits_short

L176show_nilpotent_orbits_short(RealForm G) = void
not really necessary

also in this file at line 64, line 90, line 370, line 462, line 493

show_nilpotent_orbits

L177show_nilpotent_orbits(RealForm G) = void

also in this file at line 76, line 89, line 371, line 477, line 492

show_nilpotent_orbits_long

L180show_nilpotent_orbits_long(RealForm G) = void

also in this file at line 92, line 108, line 114, line 144, line 308, line 338, line 374, line 376, line 495, line 512, line 515

nilpotent_orbits_data_long

L182nilpotent_orbits_data_long(InnerClass ic)

nilpotent_orbits_data_very_long

L222nilpotent_orbits_data_very_long(InnerClass ic_dual) = (InnerClass,[(ComplexNilpotent,[int],int,RootDatum,RootDatum,int,int,[int],[int],[ComplexNilpotent_d])])
very long: includes Arthur parameter information
the given InnerClass ic should be on the dual side,
the orbits are on the dual side, as are the counts of Arthur parameters
assert these two numbers agree:
 two ways of counting number of Arthur parameters
 arthur_parameters@(RootDatum,ComplexNilpotent) see arthur_parameters.at
 same as number of nilpotent orbits of integrality datum (dual side) with
  correct central invariant

show_nilpotent_orbits_very_long 2 overloads

L252show_nilpotent_orbits_very_long((InnerClass ic,[(ComplexNilpotent,[int],int,RootDatum,RootDatum,int,int,[int],[int],[ComplexNilpotent_d])] data)) = void
L280show_nilpotent_orbits_very_long(InnerClass ic) = void
Commented-out code, lines 283–302 (18 lines)
set show_nilpotent_orbits_very_long_plus([ComplexNilpotent] orbits,[[ComplexNilpotent_d]] arthur_parameters)=
tabulate(
["i","H","diagram","dim","BC Levi","Cent","Z","C_2","A(O)","Ar"]#for  orbit@i in orbits do
let d=orbit.component_datum then
cent=orbit.centralizer in
[i.to_string,
([int]:orbit.H).to_string,
orbit.diagram.to_string,
orbit.dim_nilpotent.to_string,
orbit.Bala_Carter_Levi.nice_format,
cent.nice_format,
cent.derived.order_center.to_string,
(#cent.compact_form.strong_real_forms_same_type).to_string,
d.orders.to_string,
(#arthur_parameters[i]).to_string
] od);
let (diagram,dim,levi,cent,Z,C_2,A_O,real_orbits)=data in
(diagram,dim,levi,cent,Z,C_2,A_O,real_orbits,arthur_parameters)

show commands for orbits of RealForm G L304

includes: number of real forms of the orbit
L305

show_nilpotent_orbits_long 2 overloads

L308show_nilpotent_orbits_long([ComplexNilpotent] orbits,RealForm G) = void
compact listing: i H diagram dim BC-Levi Cent A(O) #RF C_2
L338show_nilpotent_orbits_long([ComplexNilpotent] orbits,InnerClass ic) = void

also in this file at line 92, line 108, line 114, line 144, line 180, line 374, line 376, line 495, line 512, line 515

show_nilpotent_orbits_short

L370show_nilpotent_orbits_short(RealForm G) = void
not really necessary

also in this file at line 64, line 90, line 176, line 462, line 493

show_nilpotent_orbits

L371show_nilpotent_orbits(RealForm G) = void

also in this file at line 76, line 89, line 177, line 477, line 492

show_nilpotent_orbits_long 2 overloads

L374show_nilpotent_orbits_long(RealForm G) = void
L376show_nilpotent_orbits_long(InnerClass ic) = void

also in this file at line 92, line 108, line 114, line 144, line 180, line 308, line 338, line 495, line 512, line 515

show_nilpotent_orbits_long_plus

L419show_nilpotent_orbits_long_plus(InnerClass ic) = [([int],int,RootDatum,RootDatum,int,int,[int],[int])]
also return everything you calculate:
for each orbit:
(diagram,dim,BC Levi,cent,Z(cent_0),C_2,orders of classes of A(O), #real orbits for all forms in inner class)
(

tables of nilpotents from Springer tables L459

show_nilpotent_orbits_short

L462show_nilpotent_orbits_short(SpringerTable st) = void
SpringerTable versions: also include information about duality

also in this file at line 64, line 90, line 176, line 370, line 493

show_nilpotent_orbits 2 overloads

L477show_nilpotent_orbits(SpringerTable st) = void
include special_star, dual_orbit
L492show_nilpotent_orbits(RootDatum rd) = void

also in this file at line 76, line 89, line 177, line 371

show_nilpotent_orbits_short

L493show_nilpotent_orbits_short(RootDatum rd) = void

also in this file at line 64, line 90, line 176, line 370, line 462

show_nilpotent_orbits_long 3 overloads

L495show_nilpotent_orbits_long([ComplexNilpotent] orbits) = void
L512show_nilpotent_orbits_long(RootDatum rd) = void
L515show_nilpotent_orbits_long(SpringerTable st) = void

also in this file at line 92, line 108, line 114, line 144, line 180, line 308, line 338, line 374, line 376

show_real_nilpotent_orbits

L542show_real_nilpotent_orbits(RealForm G) = void
simple table of real nilpotent orbits

Generated from atlas-scripts at commit 648506c (2026-10-13).