Atlas of Lie Groups and Representations

Documentation contents

atlas Library · Script File

coherent_characters.at

character of coherent representation on parameters, KGB, and involutions
Source
atlas-scripts/coherent_characters.at (377 lines)
Definitions
62
Loads
Loaded by
four.at

Definitions in source order

coherent_verbose

L7coherent_verbose = true

involutions 2 overloads

L9involutions([WeylElt] W) = [WeylElt]
L12involutions(RootDatum rd) = [WeylElt]

twisted_involutions

L14twisted_involutions(RealForm G) = [mat]

epsilon_i 4 overloads

L22epsilon_i(WeylElt involution, WeylElt w) = int
sign function (not character) on W coming from sgn
character of imaginary roots
L28epsilon_i(KGBElt x,WeylElt w) = int
L31epsilon_i(Param p,WeylElt w) = int
L33epsilon_i(mat inv,WeylElt w) = int

epsilon_r 2 overloads

L37epsilon_r(KGBElt x,WeylElt w) = int
L41epsilon_r(mat inv,WeylElt w) = int

cross

L48cross (WeylElt w,Param p) = Param
FIX cross@(WeylElt,Param): wrong assert, w.root_datum should be the integrality
datum for p

also defined in basic.at, coherent_irreducible.at, synthetic.at, extended_misc.at

fixed_params

L53fixed_params(WeylElt w,[Param] params) = [Param]
fixed points of w acting on parameters

trace_coherent

L61trace_coherent(WeylElt w,[Param] params) = int
character of coherent continuation representation on block,
computed in basis of standard modules, by grading the block
representation by Cartan
w.root_datum must equal params[0].root_datum

also in this file at line 88, line 122, line 125

make_block

L65make_block (RealForm G, RealForm Gd) = [Param]

character_coherent

L69character_coherent(CharacterTable ct,[Param] params) = [int]

also in this file at line 94, line 141, line 144

character_coherent_gamma 2 overloads

L75character_coherent_gamma(RealForm G,CharacterTable ct,ratvec gamma) = [int]
L77character_coherent_gamma(RealForm G,CharacterTable ct) = [int]

character_coherent_B 2 overloads

L79character_coherent_B(RealForm G,CharacterTable ct) = [int]
L81character_coherent_B(RealForm G,RealForm Gd,CharacterTable ct) = [int]

fixed_kgb

L85fixed_kgb(WeylElt w,[KGBElt] X) = [KGBElt]
fixed points of w acting on KGB

trace_coherent

L88trace_coherent(WeylElt w,[KGBElt] X) = int

also in this file at line 61, line 122, line 125

trace_coherent_r

L91trace_coherent_r(WeylElt w,[KGBElt] X) = int

also in this file at line 128

character_coherent

L94character_coherent(CharacterTable ct,[KGBElt] X) = [int]

also in this file at line 69, line 141, line 144

character_coherent_X

L96character_coherent_X(RealForm G,CharacterTable ct) = [int]

character_coherent_r

L98character_coherent_r(CharacterTable ct,[KGBElt] X) = [int]

also in this file at line 147

character_coherent_r_X

L101character_coherent_r_X(RealForm G,CharacterTable ct) = [int]

character_coherent_dual

L104character_coherent_dual(CharacterTable ct,[KGBElt] Y) = [int]

also in this file at line 131

character_coherent_Y 2 overloads

L109character_coherent_Y(CharacterTable ct, RealForm G_dual)
L112character_coherent_Y(RealForm G,CharacterTable ct) = [int]

fixed_involutions 2 overloads

L116fixed_involutions(WeylElt w,[WeylElt] involutions) = [WeylElt]
involutions in W fixed by conjugation action of w
L119fixed_involutions(WeylElt w, [mat] involutions) = [mat]

trace_coherent 2 overloads

L122trace_coherent(WeylElt w,[WeylElt] involutions)
L125trace_coherent(WeylElt w,[mat] involutions)

also in this file at line 61, line 88

trace_coherent_r

L128trace_coherent_r(WeylElt w,[mat] involutions)

also in this file at line 91

character_coherent_dual

L131character_coherent_dual(CharacterTable ct,RealForm G_dual) = [int]

also in this file at line 104

character_coherent_dual_r

L136character_coherent_dual_r(CharacterTable ct,RealForm G_dual) = [int]

character_coherent 2 overloads

L141character_coherent(CharacterTable ct,[WeylElt] involutions)
L144character_coherent(CharacterTable ct,[mat] involutions)

also in this file at line 69, line 94

character_coherent_r

L147character_coherent_r(CharacterTable ct,[mat] involutions)

also in this file at line 98

character_coherent_T

L151character_coherent_T(RealForm G, CharacterTable ct) = [int]

character_coherent_T_r

L154character_coherent_T_r(RealForm G, CharacterTable ct) = [int]

character_coherent_T_dual

L158character_coherent_T_dual(CharacterTable ct,RealForm G_dual) = [int]

character_coherent_T_dual_r

L161character_coherent_T_dual_r(CharacterTable ct,RealForm G_dual) = [int]

character_coherent_I

L165character_coherent_I(CharacterTable ct) = [int]

coherent_data 2 overloads

L169coherent_data(SpringerTable st,[Param] block) = (vec,vec,vec)
L181coherent_data(SpringerTable st,RealForm G) = (vec,vec,vec)

show_coherent_data

L183show_coherent_data(SpringerTable st,(vec a,vec b,vec c)) = void

also in this file at line 229, line 285, line 365, line 369, line 371

show_coherent_data_short

L206show_coherent_data_short(SpringerTable st,(vec a,vec b,vec c))

show_coherent_data 2 overloads

L229show_coherent_data = (SpringerTable,RealForm->)
L285show_coherent_data (RealForm G) = void

also in this file at line 183, line 365, line 369, line 371

Lusztig_cell

L301Lusztig_cell(SpringerTable st_dual,ComplexNilpotent O) = [[int]]
see Barbasch-Trapa, Stable Combinations...Proposition 3.2
That says: V(O_check)\otimes sgn is the sum of Springer representations
 in the special piece of dual(O_check), where V(O_check) is the Lusztig
 cell consisting of all Springer(x,1)
Change roles of G,G_check, move sgn to the other side:
 rd=given RootDatum
 V(O)=\sum Springer(dual_orbit)\otimes sgn
  where dual_orbit runs over the special piece of dual(O),
   which is an array of orbits for dual(rd)
  this is an array of characters for W(dual(rd))=W(rd)
Note that we need the Springer table for dual(rd) (not rd itself)
 since the Lusztig cell is in terms of the Springer correspondence for dual(rd)
array of characters for W(rd)

Lusztig_cells

L314Lusztig_cells(SpringerTable st_dual) = [(ComplexNilpotent,[[int]])]

show_Lusztig_cells

L317show_Lusztig_cells(CharacterTable ct,[(ComplexNilpotent,[[int]])] data) = void

size_weak_packet_int

L330size_weak_packet_int(ComplexNilpotent O_check,SpringerTable st_G,[Param] params)
see assem.pdf
size of weak packet \Pi(O_check) is:
dom_hom(\sum_{O'}Springer(O'),M_gamma) where M_gamma is the coherent continuation representation on
representations of infinitesimal character gamma, and the sum runs over orbits O' in the special
piece of d(O_check)

size_weak_packet

L344size_weak_packet(ComplexNilpotent O_check,SpringerTable st_G,[Param] params)

sizes_weak_packets 3 overloads

L352sizes_weak_packets([ComplexNilpotent] orbits,SpringerTable st_G, RealForm G)
L359sizes_weak_packets(SpringerTable st_G, RealForm G)
L362sizes_weak_packets (RealForm G)

show_coherent_data 3 overloads

L365show_coherent_data(SpringerTable st,[Param] block)
L369show_coherent_data(RealForm G)
set show_coherent_data(SpringerTable st,RealForm G)=show_coherent_data(st,block_of(G.trivial))
L371show_coherent_data([RealForm] groups)

also in this file at line 183, line 229, line 285

groups

L377groups = [SL(2,R),Sp(4,R),G2_s, Sp(6,R),Sp(8,R),F4_s,SL(3,R),SL(4,R),E6_q]

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