Atlas of Lie Groups and Representations

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atlas Library · Script File

K_type_formula.at

Source
atlas-scripts/K_type_formula.at (105 lines)
Definitions
9
Loads
Loaded by
hodge_K_type_formula.at

Definitions in source order

K_type_formula_verbose

L4K_type_formula_verbose = true
first: implement the built-in K_type_formula@KType in scripts
this is K_type_formula_plus
then: modify it to work on the level of Hodge filtrations
L6
formula for K-type as alternating sum of representations restricted to K
 each term is standard, final, normal, nonzero, tempered (see standardrepk.h)
Based on KHatHowe Theorem 16.6
This is based in turn on Theorem 16.5 (Zuckerman's theorem)
Theorem 16.5 writes a one-dimensional as linear combination of standards
 use character_formula_one_dimensional for this
 (for testing: can use character_formula instead, which uses KLV polynomials)
Given a K-type mu, find (P,mu_L) as in Theorem 16.6
 Guess: (P,p_L)=reduce_good_range(parameter(mu)) (see induction.at)
Take this formula on L, with shifts from u\cap s, and cohomologically induce
The resulting parameter for G might not be standard (negative on imaginary root),
or final (negative on a complex root)
and atlas automatically standardizes/finalizes it.
For application to Hodge filtrations, print a warning when this happens
L10

u_cap_s_roots

L26u_cap_s_roots(Parabolic P,KGBElt x) = [vec]

modified_cardinality

L33modified_cardinality(KGBElt x,[vec] roots)
cardinality of set of roots, except that if alpha is complex and
|<theta_x(alpha),alpha^v>=1| then counts twice

u_cap_s_rho_shifts_and_cardinalities

L38u_cap_s_rho_shifts_and_cardinalities(Parabolic P,KGBElt x) = [(vec,int)]
each subset S of u\cap s roots -> (sum of roots in S, cardinality(S))
(null(P.root_datum.rank),0)#for A in delete(generate_all_subsets(u_cap_s_roots(P,x)),0) do (sum(A),#A) od
L43

r

L45r(RealForm G) = int

character_formula_trivial

L48character_formula_trivial(RealForm G)
Zuckerman's character formula for the trivial representation

character_formula_one_dimensional

L59character_formula_one_dimensional(Param p)
Zuckerman's character formula, adapted to any one-dimensional representation

theta_induce_standard_warning

L68theta_induce_standard_warning (Param p_L,RealForm G,bool warnings_on) = Param
copied from induction.at and modified to *not* do the standardization/finalization step

K_type_formula_plus

L82K_type_formula_plus(KType mu) = ParamPol
KType mu ->  mu=\sum_i a_i q_i  (q_i final tempered, restricted to K)
same as built-in K_type_formula, done in atlas to allow
hodge generalization

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