# aql.at Function References¶

## inf_chars_dom_for_L¶

inf_chars_dom_for_L:Param p, Parabolic P->[ratvec] Defined in line number 9.

Given a parameter p for G and a real parabolic P, list all Weyl(G) conjugates of the infinitesimal character of p that are dominant for L.

## inf_chars_for_L¶

inf_chars_for_L:Param p,Parabolic P->[ratvec] Defined in line number 18.

Given a parameter p for G and a theta-stable parabolic P, list all infinitesimal characters v for L so that v+rho(u) is the infinitesimal character of p.

## inf_chars_for_L¶

inf_chars_for_L:ratvec gamma,Parabolic P->[ratvec] Defined in line number 28.

Given an infinitesimal character gamma for G and a theta-stable parabolic P, list all infinitesimal characters v for L so that v+rho(u) is the infinitesimal character of p.

## one_dim_params_gamma¶

one_dim_params_gamma:ratvec ic, Parabolic P->[Param] Defined in line number 38.

List all one-dimensional unitary characters with given infinitesimal character.

## wf_one_dim_params¶

wf_one_dim_params:ratvec ic, Parabolic P->[Param] Defined in line number 48.

List all one-dimensional unitary characters, in the weakly fair range, of L, with given infinitesimal character.

## wf_aqs_param_pol¶

wf_aqs_param_pol:Param p, Parabolic P->[(Param,ParamPol)] Defined in line number 55.

Auxiliary function: List of all unitary weakly fair Aq(lambda) modules with infinitesimal character of p, and induced from P.

## wf_aqs_param¶

wf_aqs_param:Param p, Parabolic P->[(Param,Param)] Defined in line number 67.

Auxiliary function: As previous function, except a list of all parameters occurring as constitutents of such modules.

## is_weakly_fair_Aq_from_P¶

is_weakly_fair_Aq_from_P:Param p, Parabolic P->bool Defined in line number 77.

Decide whether p is the parameter of a (constituent of a) unitary weakly fair Aq(lambda) induced from parabolic P.

## special_theta_stable_parabolics¶

special_theta_stable_parabolics:RealForm G->[Parabolic] Defined in line number 82.

List all proper theta-stable parabolics for G that are not Borels.

## all_wf_Aq_with_ic_of¶

all_wf_Aq_with_ic_of:Param p->[Param] Defined in line number 90.

List all parameters of constituents of weakly fair Aq(lambda) modules with the same infinitesimal character as p.

## is_weakly_fair_Aq¶

is_weakly_fair_Aq:Param p->bool Defined in line number 98.

Determine whether parameter p is that of a (constituent of a) unitary weakly fair Aq(lambda) module.

## is_wf_induced_from_one_dim¶

is_wf_induced_from_one_dim:Param p->[(Parabolic,Param)] Defined in line number 114.

List all one-dimensional unitary parameters pL so that p is theta-induced from pL in the weakly fair range.

## one_dim_real_induced_param_pol¶

one_dim_real_induced_param_pol:Param p, Parabolic P->[(Param,ParamPol)] Defined in line number 137.

Auxiliary function: List of all modules with infinitesimal character of p, that are induced from a unitary character on the Levi of P.

## one_dim_real_induced_param¶

one_dim_real_induced_param:Param p, Parabolic P->[(Param,Param)] Defined in line number 149.

Auxiliary function: As previous function, except a list of all parameters occurring as constitutents of such modules.

## is_real_induced_from_character_from_P¶

is_real_induced_from_character_from_P:Param p, Parabolic P->bool Defined in line number 158.

Decide whether p is the parameter of a (constituent of a) module induced from a unitary character on the real parabolic P.

## is_real_induced_from_one_dimensional¶

is_real_induced_from_one_dimensional:Param p->bool Defined in line number 164.

Determine whether parameter p is that of a (constituent of a) module (real) induced from a unitary character.

## real_induced_from_one_dim¶

real_induced_from_one_dim:Param p->[(Parabolic,Param)] Defined in line number 175.

List all one-dimensional unitary parameters pL so that p is real-induced from pL

## wf_aqs_param_pol¶

wf_aqs_param_pol:ratvec gamma, Parabolic P->[(Param,ParamPol)] Defined in line number 187.

List of all unitary weakly fair Aq(lambda) modules with infinitesimal character gamma, and induced from P.

## wf_aqs_param¶

wf_aqs_param:ratvec gamma, Parabolic P->[(Param,Param)] Defined in line number 199.

As previous function, except a list of all parameters occurring as constitutents of such modules.

## is_unitary_by_cases¶

is_unitary_by_cases:Param p->bool Defined in line number 211.

Test whether the irreducible given by a parameter is unitary; if strongly regular, then check if good Aq. Otherwise, check whether it is real or theta induced from a unitary character; if not, compute the hermitian form.

## is_unitary_sr¶

is_unitary_sr:Param p->bool Defined in line number 229.

Test whether a representation is unitary, checking first whether it is strongly regular.

## is_unitary_reduced_with_form¶

is_unitary_reduced_with_form:Param p->(ParamPol,bool) Defined in line number 240.

Test whether a representation is unitary, checking first whether it is strongly regular; if not, reducing it in the (weakly) good range, and inducing the hermitian form of the smaller group. This function the hermitian form and a boolian.

## is_unitary_reduced¶

is_unitary_reduced:Param p->bool Defined in line number 260.

As previous function, but only returns true/false.