Atlas of Lie Groups and Representations

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Script File

G2_unitary_dual.at

Source
atlas-scripts/G2_unitary_dual.at (149 lines)
Definitions
25
Loads
Loaded by
none of the other all.at files

Definitions in source order

G

L7G = G2_s

my_simple_roots

L8my_simple_roots = mat

short_format

L10short_format(Param p) = string

all_parameters_x_gamma

L15all_parameters_x_gamma (KGBElt x,ratvec gamma) = [Param]
need to modify all_parameters_x_gamma so it doesn't make gamma dominant
all parameters with given infinitesimal character and attached to given x
keep only 1 from each equivalence class; NOTE: this finalizes each parameter
L26

M

L28M = mat

i_M

L29i_M = ratmat
M: 3x2 integral matrix, takes Z^2 -> Z^3, atlas roots -> my_simple_roots
i_M: 2x3 rational matrix, Q^2 -> Q^3, my_simple_roots -> atlas roots

i_M*M= 2x2 identity  (M*i_M \ne 3x3 identity

atlas> for a in G.posroots do prints(a, " ", M*a) od
atlas      Vogan
[  2, -1 ] [  1, -1,  0 ]    short simple
[ -3,  2 ] [ -1,  2, -1 ]    long simple
[  3, -1 ] [  2, -1, -1 ]    l
[ 0, 1 ] [  1,  1, -2 ]      l
[ -1,  1 ] [  0,  1, -1 ]    s
[ 1, 0 ] [  1,  0, -1 ]      s
L31

g2_parameter

L48g2_parameter(KGBElt x, ratvec lambda, ratvec nu)

cartans

L49cartans = Cartan_classes(G)

H1

L52H1 = cartans[1]
use numbering from [voganG2]
has short real roots

also defined in galois.at

H2

L53H2 = cartans[2]
has long real roots

H_s

L54H_s = H1

H_l

L55H_l = H2

ps

L58ps(int epsilon,ratvec v)
principal series

gamma_s

L62gamma_s(rat m,rat v) = ratvec
short root Cartan
infinitesimal character, not necessarily dominant:

x_s

L66x_s = KGB(G,4)
gamma_s=[v,(m-v)/2] (atlas)
=[m+v,m-v,-m]/2 (Vogan 9.2a)
short simple root    [2,-1]=[1,-1,0] is real
orthogonal long root [0,1] =[1,1,-2] is nci
the coroot of the nci root is [1,2]: this must be integral on gamma
[1,2]*gamma_s(m,v)=m so this is a valid gamma for x_s
L68

p_s

L74p_s(rat m, rat v) = [Param]
allow [Param] for limits of DS  p_s(even,0)

gamma_l

L81gamma_l(rat m,rat v) = ratvec
long root Cartan
infinitesimal character, not necessarily dominant:

x_l

L86x_l = KGB(G,3)
note: [3,-2] = -[-3,2] is a negative root = [1,-2,1]
gamma_l=[(m+3v)/2,-v] (atlas)
=[(m+v)/2,-v,(-m+v)/2] (10.2a)
root [-3,2] is long, simple, real
short root [1,0] = [1,0,-1] is nci (not simple)
the coroot of the nci root is [2,3]: this must be integral on gamma
[2,3]*gamma_l(m,v)=m so this is a valid gamma for x_l
L88

coords

L94coords(RootDatum G,ratvec v) = [rat]

in_fpp 3 overloads

L96in_fpp(ratvec v) = bool
L97in_fpp(RootDatum G,ratvec v) = bool
L98in_fpp(Param p) = bool

p_l

L100p_l(rat m, rat v) = [Param]
allow [Param] for limits of DS  p_s(even,0)

test_s

L105test_s(int m, rat v0, rat v1, rat step_size)

test_l

L127test_l(int m, rat v0, rat v1, rat step_size)

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