atlas Library · Script File

`dual.at`

some similar stuff to Vogan-dual.at, also some extra functionality

Source

atlas-scripts/dual.at (66 lines)

Definitions

Loads

<basic.at
<parameters.at { for parameter@(KGBElt,KGBElt,ratvec) }
<synthetic.at { for type KGBElt_gen }
<Vogan-dual.at { for |dual@(Param,InnerClass)| }
<polynomial.at { types |i_poly|, |i_poly_mat| }
<extended.at { for type ExtParam }
<KL_polynomial_matrices.at

Loaded by

jantzen.at stable.at

Definitions in source order

dual of a block L11

`dual_block`

L19dual_block ([Param] B, InnerClass dual_inner_class) = ([Param],[int])

various functions for constructing dual block:
dual(B), dual(B,g), dual(B,g,dual_inner_class), dual(B,dual_inner_class)
each returns ([Param] B_vee,[int] permutation)
B_vee is block for for G^vee(infinitesimal character)
Vogan duality is B[i] -> B_vee(permutation[i])

also in this file at line 31

B_vee is in length *increasing* order
B_dual is in length *decreasing* order; B[i]->B_dual[i] is Vogan duality
B_vee=B_dual up to permutation, roughly reversing order

L25

`dual_block`

L31dual_block ([Param] B) = ([Param],[int])

don't specify the inner class and/or the dual infinitesimal character

also in this file at line 19

`dual_ExtParameter`

L34dual_ExtParameter ( ExtParam(ic,delta,gamma,lambda,theta,g,l,omega,tau,t)) = ExtParam

KL_Q_polynomials@Param is defined in KL_polynomial_matrices

L38

`KL_Q_polynomial`

L44KL_Q_polynomial ([Param] B, i_poly_mat Q,Param irr,Param std) = i_poly

individual Q polynomials
given B to make sure we have the right basis in the right order
given Q so don't need to recompute it

`test_duality`

L55test_duality (Param p) = (i_poly_mat,i_poly_mat,bool)

in some cases this test fails because it is off by a permutation which
I don't understand

for example G=Sp(6,R), test_duality(spherical_principal_series(G,rho(G)))

returns P,Q, and true/false;
P*Q should be the identity

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