Atlas of Lie Groups and Representations

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K_types_plus.at

Source
atlas-scripts/K_types_plus.at (141 lines)
Definitions
12
Loads
Loaded by
K_type_formula.at

Definitions in source order

find

L4find ([KType] ktypes,KType mu) = int

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

Enhanced version of K_types@KHighestWeight (in K_highest_weights.at)
Also the algorithm \tau->tau_q in KHatHowe Theorem 11.9
L6
algorithm
 given K_type p (STFL for G)
 find (Q,p_L),
   Q=theta-stable parabolic
   p_L=parameter for Q (NOT STFL)
 AND p_L is the parameter of a one-dimensional representation of L
 such that p=theta_induce_standard(p_L*0,G)
 Q is the "strongly compatible parabolic" of KHatHowe 11.6
 p_L is a minimal principal series of L,
  J(p_L) is one-dimensional of L, whose restriction to L\cap K
  is denoted tau_K in KHatHowe Theorem 16.6
The algorithm here is crazily roundabout, using
stuff from K_highest_weights.at
Algorithm:
given a KType mu
compute a single highest weight lambda of mu
then K_types(lambda) returns a set of K-types of G
with this highest weight, among them mu itself
furthermore it realizes each of these K-types
as induced from Q
so K_types_plus@KHighestWeight is the same as
K_types@KHighestWeight  (see K_highest_weights.at)
except it keeps around extra information: Q
and an array [(p_L_i,mu_i)]
then run through the p_L's and pick the one which
actually gives mu_i=mu
L9

inds_plus

L38inds_plus (KHighestWeight mu_in) = [(KType,[Param])]
Auxiliary function (see inds in K_highest_weights.at for some details).

choose_irred_inds_plus

L48choose_irred_inds_plus ([[(KType,[Param])]] list) = [[(KType,[Param])]]

K_types_plus 2 overloads

L57K_types_plus (KHighestWeight(,x,v):mu_in) = (Parabolic,[(Param,KType)])
The same as K_types@KHighestWeight, but return more information:
the associated theta-stable parabolic, and for each K_type, also the
L-spherical parameter on L
L75K_types_plus (KType mu) = (Parabolic,[(Param,KType)])

test_K_types_plus 2 overloads

L78test_K_types_plus(KHighestWeight lambda) = bool
L86test_K_types_plus([KHighestWeight] lambdas) = bool

tau_q

L96tau_q(KType mu) = (Parabolic,Param)
K_type p -> (Q,p_L) such that
 p=theta_induce_standard(p_L*0,G)
 Note: p_L*0 not p_L
 and J(p_L) is one-dimensional
See KHatHowe Theorems 11.9 and 16.6

cross_divide_alt

L105cross_divide_alt (KGBElt x,KGBElt y) = (bool,WeylElt)

is_strongly_compatible

L114is_strongly_compatible (Param p,Parabolic P) = bool

make_strongly_compatible_parabolic

L131make_strongly_compatible_parabolic (Param p) = Parabolic

test_parabolics

L137test_parabolics (RealForm G) = bool

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