Atlas of Lie Groups and Representations

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associated_variety_annihilator.at

Source
atlas-scripts/associated_variety_annihilator.at (455 lines)
Definitions
37
Loads
Loaded by
none of the other all.at files

Definitions in source order

av_verbose

L5av_verbose = false
Commented-out code, lines 7–15 (7 lines)
set restrict_character_long(RootDatum rd ,RootDatum L,CharacterTable ct_rd,CharacterTable ct_int,[int] char)=
let ()=prints("RESTRICTING: ",new_line,
"rd=", rd, rd.simple_roots,new_line,
"L=", L, L.simple_roots,new_line,
"ct_rd=", ct_rd.root_datum, ct_rd.root_datum.simple_roots,new_line,
"ct_int=", ct_int.root_datum, ct_int.root_datum.simple_roots) in
restrict_character(rd,L,ct_rd,ct_int,char)
SEVERAL VERSIONS
associated_variety_ann_old should be ignored: this only works if some permutation is the identity
associated_variety_ann_inid includes the permutation,
  and uses truncated induction to go from the integrality_datum to G
associated_variety_ann_res
  similar, but uses restriction instead of truncated induction, using:
  ind_{W(L)}^{W(G)}(sigma_L)= unique irreducible of W(G) with the same degree (b-value)
    containing sigma_L in the restriction
associated_variety_ann is an alias to associated_variety_ann_res
L17
old versions have been deleted
L28

associated_variety_ann_ind 6 overloads

L33associated_variety_ann_ind ( Param param , SpringerTable st_rd , CharacterTable ct_int , WCell cell ) = ComplexNilpotent
using truncated induction, incorporating permutation
only need Springer table for G, character table for L=integrality datum
new: includes permutation
L64associated_variety_ann_ind (Param p,SpringerTable st_rd,WCell cell) = ComplexNilpotent
given cell, SpringerTable for G, compute ct_int, using restriction
L69associated_variety_ann_ind (Param p,WCell cell) = ComplexNilpotent
(slow) given cell, compute st_G,ct_int
L74associated_variety_ann_ind(Param p,SpringerTable st_G,CharacterTable ct_int) = ComplexNilpotent
given Param, SpringerTable for G and CharacterTable for integrality_datum,compute W_cell
L77associated_variety_ann_ind(Param p,SpringerTable st_G) = ComplexNilpotent
L81associated_variety_ann_ind(Param p) = ComplexNilpotent

using restriction L84

associated_variety_int 4 overloads

L87associated_variety_int (SpringerTable st,WCell cell) = ComplexNilpotent
assuming integral infinitesimal character, no induction
L89associated_variety_int(SpringerTable st) = (WCell->ComplexNilpotent)
L90associated_variety_int(SpringerTable st,Param p) = ComplexNilpotent
L91associated_variety_int(SpringerTable st) = (Param->ComplexNilpotent)

associated_variety_ann_res 6 overloads

L95associated_variety_ann_res (SpringerTable st_rd, CharacterTable ct_int, WCell cell) = ComplexNilpotent
given cell, SpringerTable for G and CharacterTable for integrality_datum,
using restriction
L133associated_variety_ann_res(SpringerTable st_rd,WCell cell) = ComplexNilpotent
given cell, SpringerTable for G, compute ct_L, using restriction
L138associated_variety_ann_res(RootDatum rd,WCell cell) = ComplexNilpotent
(slow) given cell, compute st_rd,ct_L
L144associated_variety_ann_res (Param p,SpringerTable st_rd,CharacterTable ct_L) = ComplexNilpotent
given Param, SpringerTable for rd and CharacterTable for integrality_datum,
compute W_cell
L148associated_variety_ann_res(Param p,SpringerTable st_rd) = ComplexNilpotent
L152associated_variety_ann_res(Param p) = ComplexNilpotent

associated_variety_ann_res_test

L155associated_variety_ann_res_test(SpringerTable st_rd,WCell cell)
chars_rd=
##for i:st_rd.ct.n_irreps do
 if ct.degree(i)=d_L and
  !=ct_L.inner_product(char_L,restrict_character(rd,L,ct_rd,ct_L,ct.characters[i]))
  then  [i] else [] fi
od
then
index_rd=chars_rd[0] in
st_rd.springer_inverse(index_rd)
L164

associated_variety_ann 5 overloads

L177associated_variety_ann = (SpringerTable,CharacterTable,WCell->ComplexNilpotent)
some aliases
L181associated_variety_ann = (SpringerTable,WCell->ComplexNilpotent)
L184associated_variety_ann = (RootDatum,WCell->ComplexNilpotent)
L186associated_variety_ann = (Param,SpringerTable,CharacterTable->ComplexNilpotent)
L189associated_variety_ann = (Param->ComplexNilpotent)
Commented-out code, lines 192–192 (1 lines)
set GK_dim(Param p,SpringerTable st)=int:associated_variety_ann(p,st).dim_nilpotent\2

GK_dim

L195GK_dim(Param p) = int
If you only want the GK dimension you don'tn need to compute the induction step.
Commented-out code, lines 200–200 (1 lines)
set GK_dim(SpringerTable st)=(Param->int):(Param p):GK_dim(p,st)

show

L203show([(Param,ComplexNilpotent)] data) = void

also defined in modules.at, K_Nilpotent.at, good_W_representatives.at, arthur_parameters.at, sub_cells.at, geck_generic.at, L_packet.at

associated_variety_ann_res_complex

L211associated_variety_ann_res_complex(Param p) = ComplexNilpotent
G is a complex group; use left cell
rd=cell.root_datum
   then st_rd=Springer_table(rd)
   then ct_int=p.integrality_datum.character_table
   then L=cell.root_datum
   then index_pi_int=ct_int.special_character_inefficient(cell)
   then char_int=ct_int.characters[index_pi_int]
   then d_int=ct_int.degree(index_pi_int)
   then chars_rd=
##for i:st_rd.ct.n_irreps do
{ let ()=prints("i: ",i, " ",  ct_int.degree(i), " ", d_int) in }
 if ct_int.degree(i)=d_int and
  !=ct_int.inner_product(char_int,restrict_character(rd,L,ct_int,ct_int,ct_int.characters[i]))
  then  [i] else [] fi
od
then
index_rd=chars_rd[0]
   then orbit=st_rd.springer_inverse(index_rd)
in orbit
L216

av_maximal_ideal_0 3 overloads

L239av_maximal_ideal_0(RootDatum G,ratvec gamma,SpringerTable st,CharacterTable ct_int)
G: real group, st: SpringerTable for GC,ct_int: CharacterTable for GC,gamma#gamma
this is slow, see av_maximal_ideal for much faster version
L249av_maximal_ideal_0(RootDatum rd,ratvec gamma,SpringerTable st_C)
L252av_maximal_ideal_0 (RootDatum G,ratvec gamma)

av_maximal_ideal 4 overloads

L269av_maximal_ideal ( SpringerTable st , CharacterTable ct_int , CharacterTable ct_singular , ratvec gamma ) = ComplexNilpotent
much faster version:
rd_singular \subset rd_int \subset rd
d=#rd_singular.posroots
look for character sigma of W(rd_int) with b(sigma)=d
 and containing the sign representation of W(rd_singular)
tensor with sign, and truncated induce to get
 a character tau of W(rd)
return springer_inverse(tau)

precompute:
Springer_table(rd)
character_table(rd_int)
character_table(rd_singular)
L336av_maximal_ideal (SpringerTable st,CharacterTable ct_int,ratvec gamma) = ComplexNilpotent
precompute st,ct_int
L346av_maximal_ideal (SpringerTable st,ratvec gamma) = ComplexNilpotent
precompute st
L352av_maximal_ideal(RootDatum rd,ratvec gamma)

test_av_maximal_ideal 2 overloads

L361test_av_maximal_ideal(SpringerTable st) = bool
L385test_av_maximal_ideal (RootDatum rd) = bool

GK_dim_maximal_ideal 2 overloads

L390GK_dim_maximal_ideal ( RootDatum rd , CharacterTable ct_int , CharacterTable ct_singular , ratvec gamma ) = int
GK_dim_maximal_ideal is like GK_dim: it returns just the dimension of the orbit,
and can therefore skip (one of the) the induction step(s)
L449GK_dim_maximal_ideal(RootDatum rd,ratvec gamma) = int

Generated from atlas-scripts at commit 7e1b958 (2026-09-17).