Atlas of Lie Groups and Representations

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disconnected.at

data type for certain disconnected complex reductive groups

for now: class which includes G^theta where G is connected complex,
theta is an involution. G/G^theta is an elementary 2-group
Source
atlas-scripts/disconnected.at (90 lines)
Definitions
13
Loads
Loaded by
arthur_parameters.at

Definitions in source order

DisconnectedGroup type

L10
set_type [DisconnectedGroup = (RootDatum rd,[vec] characters,[mat] automorphisms)]

Fields: rd, characters, automorphisms

characters: representatives of the 2-group ker(1+delta)/im(1-delta)=X^*(H^delta/(H^delta)_0)
automorphisms: a set A of square matrices of size rank(rd)a, satisfying:
1) M^2=1  (M\in A)
2) MN=NM  (M,N in A)
3) M is a distinguished automorphism:
   M*: simple roots -> simple_roots
   (^M)*: simple_coroots -> simple_coroots

characters: see K_highest_weights.at
applying the following construction with theta=distinguished involution delta

X^*(H^theta)=X^*/(1-theta)X^*
X^*(H/(H^theta)_0)=X^*/(X^*)^{-theta}
                  =X^*/ker(1+theta)
=>
X^*(H^theta/(H^theta)_0)=(X^*)^{-theta}/(1-theta)X^*
                        =ker(1+theta)/image(1-theta)

this is from characters_of_order_2@KGBElt in K_highest_weights.at

{This function is the same as basis_lambda_differential_0 in all_parameters.at
   List of weights representative of a set of generators of the elementary
   2-group ker(1+theta(x))/im(1-theta(x)).
characters_order_2@KGBElt
returns [v_0,...,v_k], v_i\in X^*,
the characters of H^{theta_x} with differential 0
are \sum a_i v_i with a_i=0,1 (or [] -> [0,...,0])

Each v_i is in ker(1+theta)X^*
Two such give the same character <=> difference is in im(1-theta)X^*
}
L11

K

L44K(RealForm G) = DisconnectedGroup

ComplexNilpotent_d type

L51
set_type [ComplexNilpotent_d=(DisconnectedGroup rd_d,void .,vec H)]

Fields: rd_d, H

root_datum

L53root_datum(ComplexNilpotent_d O_d)

also defined in basic.at, W_reps.at, K_Nilpotent.at, sommers.at, parabolics.at, K.at, nilpotent_orbits.at, W_classes.at, extended_misc.at, character_tables.at, springer_tables.at, cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

=

L55=(ComplexNilpotent_d a,ComplexNilpotent_d b) = bool

also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, cyclotomic.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

find

L58find([ComplexNilpotent_d] orbits,ComplexNilpotent_d O) = int

also defined in basic.at, tits_centralizer.at, nilpotent_orbits.at, cells.at, K_types.at, hodgeParamLaurentPol.at, extended.at, nilpotent_posets.at, K_types_plus.at

lift

L62lift(ComplexNilpotent O,DisconnectedGroup G) = ComplexNilpotent_d
lift orbit from G_0 to G

also defined in hash.at, tits.at, center.at, nilpotent_orbits.at

saturate 2 overloads

L66saturate(ComplexNilpotent_d O_d,RootDatum rd) = ComplexNilpotent
saturate orbit of disconnected subgroup to orbit for connected G
L70saturate(ComplexNilpotent_d O_d,RootDatum rd,mat inc) = ComplexNilpotent
saturate an orbit from a disconnected subgroup H, with matrix giving X_*(T_H) -> X_*(T)

also defined in nilpotent_orbits.at, nilpotent_posets.at

remove_duplicates

L73remove_duplicates([ComplexNilpotent_d] orbits) = [ComplexNilpotent_d]

also defined in restricted_roots.at, parameters.at, nilpotent_orbits.at

intersect

L80intersect(ComplexNilpotent_d O_d) = [ComplexNilpotent]
nilpotent orbit for G intersect G_0

also defined in complex.at

nilpotent_orbits

L84nilpotent_orbits(DisconnectedGroup G) = [ComplexNilpotent_d]

also defined in nilpotent_orbits.at

orbits

L89orbits(DisconnectedGroup G) = [ComplexNilpotent_d]

also defined in nilpotent_orbits.at

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