.. _thetastable.at_ref:

thetastable.at Function References
=======================================================
|

.. _map_into_distinguished_fiber_kgbelt_x->kgbelt1:

map_into_distinguished_fiber
-------------------------------------------------
| ``map_into_distinguished_fiber:KGBElt x->KGBElt`` Defined in line number 92.
| 
| (Auxiliary function)
| 

.. _strong_map_into_distinguished_fiber_kgbelt_x->kgbelt1:

strong_map_into_distinguished_fiber
-------------------------------------------------
| ``strong_map_into_distinguished_fiber:KGBElt x->KGBElt`` Defined in line number 109.
| 
| Map KGB element x to x_K in the distinguished fiber; if necessary, use complex   cross actions first to move x to a fiber with no C- roots.
| 

.. _canonical_x_k_kgbelt_x->kgbelt1:

canonical_x_K
-------------------------------------------------
| ``canonical_x_K:KGBElt x->KGBElt`` Defined in line number 113.
| 
| Same as previous function.
| 

.. _canonical_x_k_param_p->kgbelt1:

canonical_x_K
-------------------------------------------------
| ``canonical_x_K:Param p->KGBElt`` Defined in line number 116.
| 
| Previous function with input a parameter p; it is applied to x(p).
| 

.. _u_kgbelt_x->mat1:

u
-------------------------------------------------
| ``u:KGBElt x->mat`` Defined in line number 120.
| 
| Positive coroots in the nilradical of the theta-stable parabolic determined by x.
| 

.. _rho_u_cx_parabolic_p->ratvec1:

rho_u_cx
-------------------------------------------------
| ``rho_u_cx:Parabolic P->ratvec`` Defined in line number 131.
| 
| Half sum of positive complex roots (on fundamental Cartan) in the nilradical of P;   P must be theta-stable.
| 

.. _rho_u_cx_t_parabolic_p->vec1:

rho_u_cx_T
-------------------------------------------------
| ``rho_u_cx_T:Parabolic P->vec`` Defined in line number 147.
| 
| Element of :math:`X^*`  with same restriction to :math:`(X^*)^{\theta}`  as rho_u_cx(P);   P must be theta-stable.
| 

.. _rho_u_ic_parabolic_p->ratvec1:

rho_u_ic
-------------------------------------------------
| ``rho_u_ic:Parabolic P->ratvec`` Defined in line number 157.
| 
| Half sum of imaginary compact roots in nilradical of (theta-stable) P.
| 

.. _two_rho_u_cap_k_parabolic_p->vec1:

two_rho_u_cap_k
-------------------------------------------------
| ``two_rho_u_cap_k:Parabolic P->vec`` Defined in line number 165.
| 
| Sum of compact roots (of :math:`\mathfrak t` ) in :math:`\mathfrak u`  for theta-stable parabolic P.
| 

.. _two_rho_u_cap_s_parabolic_p->vec1:

two_rho_u_cap_s
-------------------------------------------------
| ``two_rho_u_cap_s:Parabolic P->vec`` Defined in line number 169.
| 
| Sum of non-compact roots in :math:`\mathfrak u`  (for theta-stable parabolic).
| 

.. _rho_u_cap_k_parabolic_p->ratvec1:

rho_u_cap_k
-------------------------------------------------
| ``rho_u_cap_k:Parabolic P->ratvec`` Defined in line number 174.
| 
| Half sum of compact roots in :math:`\mathfrak u`  (for theta-stable parabolic).
| 

.. _rho_u_cap_s_parabolic_p->ratvec1:

rho_u_cap_s
-------------------------------------------------
| ``rho_u_cap_s:Parabolic P->ratvec`` Defined in line number 177.
| 
| Half sum of non-compact roots in :math:`\mathfrak u`  (for theta-stable parabolic).
| 

.. _dim_u_parabolic_p->int1:

dim_u
-------------------------------------------------
| ``dim_u:Parabolic P->int`` Defined in line number 180.
| 
| Dimension of :math:`\mathfrak u`  (nilrad of theta-stable parabolic).
| 

.. _dim_u_kgbelt_x->int1:

dim_u
-------------------------------------------------
| ``dim_u:KGBElt x->int`` Defined in line number 183.
| 
| Dimension of the nilradical of the theta-stable parabolic determined by KGB elt x.
| 

.. _dim_u_cap_k_parabolic_(,x):p->int1:

dim_u_cap_k
-------------------------------------------------
| ``dim_u_cap_k:Parabolic (,x):P->int`` Defined in line number 189.
| 
| Dimension of :math:`\mathfrak u\cap\mathfrak k`  for theta-stable parabolic.
| 

.. _dim_u_cap_k_kgbelt_x->int1:

dim_u_cap_k
-------------------------------------------------
| ``dim_u_cap_k:KGBElt x->int`` Defined in line number 200.
| 
| Dimension of :math:`\mathfrak u\cap\mathfrak k`  for theta-stable parabolic determined   by x.
| 

.. _dim_u_cap_k_ratvec_lambda,kgbelt_x->int1:

dim_u_cap_k
-------------------------------------------------
| ``dim_u_cap_k:ratvec lambda,KGBElt x->int`` Defined in line number 204.
| 
| Dimension of :math:`\mathfrak u\cap\mathfrak k`  for theta-stable parabolic determined by   weight lambda.
| 

.. _dim_u_cap_p_parabolic_(,x):p->int1:

dim_u_cap_p
-------------------------------------------------
| ``dim_u_cap_p:Parabolic (,x):P->int`` Defined in line number 209.
| 
| Dimension of :math:`\mathfrak u\cap\mathfrak p`  for theta-stable parabolic.
| 

.. _dim_u_cap_p_kgbelt_x->int2:

dim_u_cap_p
-------------------------------------------------
| ``dim_u_cap_p:KGBElt x->int`` Defined in line number 220.
| 
| Dimension of :math:`\mathfrak u \cap\mathfrak p`  for theta-stable parabolic associated   to x.
| 

.. _dim_u_cap_p_ratvec_lambda,kgbelt_x->int1:

dim_u_cap_p
-------------------------------------------------
| ``dim_u_cap_p:ratvec lambda,KGBElt x->int`` Defined in line number 224.
| 
| Dimension of :math:`\mathfrak u\cap\mathfrak p`  for theta-stable parabolic determined by   weight lambda.
| 

.. _dim_u_cap_k_2_parabolic_p,ratvec_h->int1:

dim_u_cap_k_2
-------------------------------------------------
| ``dim_u_cap_k_2:Parabolic P,ratvec H->int`` Defined in line number 229.
| 
| (Auxiliary function)
| 

.. _dim_u_cap_k_ge2_parabolic_p,ratvec_h->int1:

dim_u_cap_k_ge2
-------------------------------------------------
| ``dim_u_cap_k_ge2:Parabolic P,ratvec H->int`` Defined in line number 240.
| 
| (Auxiliary function)
| 

.. _dim_u_cap_p_ge2_parabolic_p,ratvec_h->int1:

dim_u_cap_p_ge2
-------------------------------------------------
| ``dim_u_cap_p_ge2:Parabolic P,ratvec H->int`` Defined in line number 251.
| 
| (Auxiliary function)
| 

.. _dim_u_cap_k_1_parabolic_p,ratvec_h->int1:

dim_u_cap_k_1
-------------------------------------------------
| ``dim_u_cap_k_1:Parabolic P,ratvec H->int`` Defined in line number 262.
| 
| (Auxiliary function)
| 

.. _make_dominant_kgbelt_x_in,ratvec_lambda_in,_ratvec_lambda_q_in->(kgbelt,ratvec,ratvec)1:

make_dominant
-------------------------------------------------
| ``make_dominant:KGBElt x_in,ratvec lambda_in, ratvec lambda_q_in->(KGBElt,ratvec,ratvec)`` Defined in line number 295.
| 
| Conjugate the triple (x,lambda, lambda_q) to make lambda_q weakly   dominant (auxiliary function).
| 

.. _aq_param_pol_kgbelt_x_in,ratvec_lambda_in,_ratvec_lambda_q->parampol1:

Aq_param_pol
-------------------------------------------------
| ``Aq_param_pol:KGBElt x_in,ratvec lambda_in, ratvec lambda_q->ParamPol`` Defined in line number 302.
| 
| A_q(lambda) module; :math:`\mathfrak q`  is defined by the weight lambda_q; x_in   must be attached to the fundamental Cartan. The module is defined as a ParamPol,   in case it is reducible.
| 

.. _aq_kgbelt_x_in,ratvec_lambda_in,_ratvec_lambda_q->param1:

Aq
-------------------------------------------------
| ``Aq:KGBElt x_in,ratvec lambda_in, ratvec lambda_q->Param`` Defined in line number 324.
| 
| A_q(lambda) module defined as above, but as a parameter, assuming it is   irreducible.
| 

.. _aq_kgbelt_x,ratvec_lambda_in->param1:

Aq
-------------------------------------------------
| ``Aq:KGBElt x,ratvec lambda_in->Param`` Defined in line number 332.
| 
| If not provided, assume lambda_q=lambda_in in the definition of A_q.
| 

.. _aq_realform_g,ratvec_lambda_in,_ratvec_lambda_q->param1:

Aq
-------------------------------------------------
| ``Aq:RealForm G,ratvec lambda_in, ratvec lambda_q->Param`` Defined in line number 336.
| 
| A_q(lambda), specify G, not x, to use x=KGB(G,0).
| 

.. _aq_realform_g,ratvec_lambda_in->param1:

Aq
-------------------------------------------------
| ``Aq:RealForm G,ratvec lambda_in->Param`` Defined in line number 340.
| 
| A_q(lambda), specify G, not x, and use lambda_q=lambda_in.
| 

.. _is_one_dimensional_param_p->bool1:

is_one_dimensional
-------------------------------------------------
| ``is_one_dimensional:Param p->bool`` Defined in line number 347.
| 
| Decide whether a parameter defines a one-dimensional representation.
| 

.. _is_good_kgbelt_x_in,ratvec_lambda_in,ratvec_lambda_q_in->bool1:

is_good
-------------------------------------------------
| ``is_good:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool`` Defined in line number 351.
| 
| Decide whether A_q(lambda) is good.
| 

.. _is_weakly_good_kgbelt_x_in,ratvec_lambda_in,ratvec_lambda_q_in->bool1:

is_weakly_good
-------------------------------------------------
| ``is_weakly_good:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool`` Defined in line number 356.
| 
| Decide whether A_q(lambda) is weakly good.
| 

.. _is_fair_kgbelt_x_in,ratvec_lambda_in,ratvec_lambda_q_in->bool1:

is_fair
-------------------------------------------------
| ``is_fair:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool`` Defined in line number 361.
| 
| Decide whether A_q(lambda) is fair.
| 

.. _is_weakly_fair_kgbelt_x_in,ratvec_lambda_in,ratvec_lambda_q_in->bool1:

is_weakly_fair
-------------------------------------------------
| ``is_weakly_fair:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool`` Defined in line number 366.
| 
| Decide whether A_q(lambda) is weakly fair.
| 

.. _goodness_kgbelt_x,ratvec_lambda_in,ratvec_lambda_q->void1:

goodness
-------------------------------------------------
| ``goodness:KGBElt x,ratvec lambda_in,ratvec lambda_q->void`` Defined in line number 372.
| 
| Determine the "goodness" of an Aq(lambda); returns "good", "weakly good",   "fair", "weakly fair", or "none".
| 

.. _is_good_param_p_l,parabolic_p,realform_g->bool1:

is_good
-------------------------------------------------
| ``is_good:Param p_L,Parabolic P,RealForm G->bool`` Defined in line number 388.
| 
| Decide whether a parameter for L is in the good range for u (only   makes sense if L is Levi of P and P is theta-stable).
| 

.. _is_weakly_good_param_p_l,parabolic_p,realform_g->bool1:

is_weakly_good
-------------------------------------------------
| ``is_weakly_good:Param p_L,Parabolic P,RealForm G->bool`` Defined in line number 394.
| 
| Decide whether a parameter for L is in the weakly good range for u   (only makes sense if L is Levi of P and P is theta-stable).
| 

.. _is_fair_param_p_l,parabolic_p,realform_g->bool1:

is_fair
-------------------------------------------------
| ``is_fair:Param p_L,Parabolic P,RealForm G->bool`` Defined in line number 400.
| 
| Decide whether a parameter for L is in the fair range for u (only   makes sense if L is Levi of P and P is theta-stable).
| 

.. _is_weakly_fair_param_p_l,parabolic_p,realform_g->bool1:

is_weakly_fair
-------------------------------------------------
| ``is_weakly_fair:Param p_L,Parabolic P,RealForm G->bool`` Defined in line number 407.
| 
| Decide whether a parameter for L is in the weakly fair range   for u (provided pi_L is one-dimensional; only   makes sense if L is Levi of P and P is theta-stable).
| 

.. _goodness_param_p_l,parabolic_p,realform_g->string1:

goodness
-------------------------------------------------
| ``goodness:Param p_L,Parabolic P,RealForm G->string`` Defined in line number 413.
| 
| Determine the "goodness" for :math:`\mathfrak u`  of a parameter for L;    returns "good", "weakly good", "fair", "weakly fair", or "none".
| 

.. _is_good_param_p_l,realform_g->bool1:

is_good
-------------------------------------------------
| ``is_good:Param p_L,RealForm G->bool`` Defined in line number 430.
| 
| Decide whether a parameter for L is in the good range for G; this only    makes sense if L is the Levi of a theta_stable parabolic.
| 

.. _is_weakly_good_param_p_l,realform_g->bool1:

is_weakly_good
-------------------------------------------------
| ``is_weakly_good:Param p_L,RealForm G->bool`` Defined in line number 441.
| 
| Decide whether a parameter for L is in the weakly good range for G; this only    makes sense if L is the Levi of a theta_stable parabolic.
| 

.. _is_fair_param_p_l,realform_g->bool1:

is_fair
-------------------------------------------------
| ``is_fair:Param p_L,RealForm G->bool`` Defined in line number 448.
| 
| Decide whether a parameter for L is in the fair range for G; this only    makes sense if L is the Levi of a theta_stable parabolic.
| 

.. _is_weakly_fair_param_p_l,realform_g->bool1:

is_weakly_fair
-------------------------------------------------
| ``is_weakly_fair:Param p_L,RealForm G->bool`` Defined in line number 460.
| 
| Decide whether a parameter for L is in the weakly fair range for G; this only    makes sense if L is the Levi of a theta_stable parabolic, and is only defined    if p_L is one-dimensional.
| 

.. _goodness_param_p_l,realform_g->void1:

goodness
-------------------------------------------------
| ``goodness:Param p_L,RealForm G->void`` Defined in line number 469.
| 
| Determine the "goodness" of a parameter for L; returns "good", "weakly good",   "fair", "weakly fair", or "none"; only makes sense if L is Levi of theta-stable   parabolic.
| 

.. _aq_packet_realform_g,complexparabolic_p->[param]1:

Aq_packet
-------------------------------------------------
| ``Aq_packet:RealForm G,ComplexParabolic P->[Param]`` Defined in line number 485.
| 
| List all A_q(0) (actually: R_q(trivial): infinitesimal character rho(G)) modules   with Q a theta-stable parabolic of type P.
| 

.. _aq_packet_realform_g,[int]_s->[param]:aq_packet(g,complexparabolic1:

Aq_packet
-------------------------------------------------
| ``Aq_packet:RealForm G,[int] S->[Param]:Aq_packet(G,ComplexParabolic`` Defined in line number 494.
| 
| List all A_q(0) (infinitesimal character rho(G)) modules   with Q a theta-stable parabolic of type S (list of simple roots).
| 

.. _aq_packet_realform_g,[*]_s->[param]:aq_packet(g,[int]1:

Aq_packet
-------------------------------------------------
| ``Aq_packet:RealForm G,[*] S->[Param]:Aq_packet(G,[int]`` Defined in line number 496.
| 
|