induction.at Function References¶
embed_KGB¶
embed_KGB:KGBElt x_L,RealForm G->KGBElt Defined in line number 89.If L is a theta-stable Levi factor in G, KGB for L embeds in KGB for G.
inverse_embed_KGB¶
inverse_embed_KGB:KGBElt x_G,RealForm L->KGBElt Defined in line number 93.Given a KGB element of G, find one for the theta-stable Levi L which maps to it.
makeS¶
makeS:mat theta,RootDatum rd->mat Defined in line number 102.Given an involution theta and a root datum, return the set S of complex roots containing the first positive representative of each quadruple ( \(\pm\) alpha, \(\pm\) theta(alpha)).
makeS¶
makeS:KGBElt x->mat Defined in line number 107.As the previous function, with argument a KGB element x determining the involution and root datum
rho_S¶
rho_S:(mat,RootDatum)pair->ratvec Defined in line number 110.Half sum of roots in chosen set S of complex roots, described above.
rho_S¶
rho_S:KGBElt x->ratvec Defined in line number 113.As previous function, with argument KGB element x.
make_parabolic¶
make_parabolic:RealForm L,RealForm G->Parabolic Defined in line number 117.Given a Levi subgroup L of G, construct the parabolic with Levi L (this reverses Levi(P) defined in parabolics.at).
real_induce_standard¶
real_induce_standard:Param p_L,RealForm G->Param Defined in line number 125.Real parabolic induction of a standard module of real Levi L (i.e., L must be the Levi factor of a real parabolic subgroup) to G
real_induce_standard¶
real_induce_standard:ParamPol P,RealForm G->ParamPol Defined in line number 136.Real parabolic induction of standards, applied to a formal sum of parameters (ParamPol).
real_induce_irreducible_as_sum_of_standards¶
real_induce_irreducible_as_sum_of_standards:Param p_L, RealForm G->ParamPol Defined in line number 142.Write the (real) induced of an irreducible J(p_L) of L as a formal sum of standards for G; uses the character formula to write J(p_L) as a formal sum of standards for L first. (Auxiliary function)
real_induce_irreducible_final¶
real_induce_irreducible_final:Param p_L, RealForm G->ParamPol Defined in line number 151.Write the (real) induced \(Ind(J(p_L))\) of an irreducible of |L| as a sum of irreducibles for |G|; uses composition series to convert output of the |real_induce_irreducible_as_sum_of_standards| into sum of irreducibles. The real form |L| of |p_L| must be the Levi factor of a real parabolic subgroup; and the parameter p_L must be final.
real_induce_irreducible¶
real_induce_irreducible:ParamPol P,RealForm G->ParamPol Defined in line number 156.Given a polynomial of parameters of L, real induce each term, and write the result as a polynomial of parameters for G.
cuspidal_data¶
cuspidal_data:Param p->(Parabolic,Param) Defined in line number 170.Cuspidal data associated to a parameter p: a cuspidal parabolic subgroup P=MN and parameter p_M for a relative limit of discrete series so that Ind(I(p_M))=I(p); uses real_parabolic(x) of parabolics.at
theta_stable_data¶
theta_stable_data:Param p->(Parabolic,Param) Defined in line number 191.Theta-stable data associated to a parameter p: a theta-stable parabolic P=LN with L relatively split, and parameter p_L for a principal series representation so that p is obtained by cohomological parabolic induction from p_L; uses theta_stable_parabolic(x) of parabolics.at.
coherent_std_imaginary¶
coherent_std_imaginary:W_word w,Param p->ParamPol Defined in line number 208.Auxiliary function
standardize¶
standardize:Param p->ParamPol Defined in line number 224.Convert a possibly non-standard parameter into a linear combination of standard ones
standardize¶
standardize:ParamPol P->ParamPol Defined in line number 235.Standardize a formal linear combination of possibly non-standard parameters
theta_induce_standard¶
theta_induce_standard:Param p_L,RealForm G->ParamPol Defined in line number 242.Theta-stable (cohomological) parabolic induction of a standard module for the Levi L of a theta-stable parabolic; if outside of weakly good range, must apply standardize.
theta_induce_parampol¶
theta_induce_parampol:ParamPol P, RealForm G->ParamPol Defined in line number 270.Given a ParamPol P, form a new ParamPol by theta-inducing each summand.
theta_induce_irreducible_as_sum_of_standards¶
theta_induce_irreducible_as_sum_of_standards:Param p_L, RealForm G->ParamPol Defined in line number 280.Write the (theta-stable) induced of an irreducible J(p_L) of L as a formal sum of standards for G; uses the character formula to write J(p_L) as a formal sum of standards for L first. (Auxiliary function)
theta_induce_irreducible_final¶
theta_induce_irreducible_final:Param p_L, RealForm G->ParamPol Defined in line number 295.Write the (theta-stable) induced Ind(J(p_L)) of an irreducible of L as a sum of irreducibles for G; uses composition series to convert output of the previous function into sum of irreducibles. The subgroup L must be the Levi factor of a theta-stable parabolic. The parameter p_L must be final.
theta_induce_irreducible¶
theta_induce_irreducible:ParamPol P,RealForm G->ParamPol Defined in line number 304.Given a polynomial of parameters, theta-induce each constituent, and write the result as a polynomial of parameters.
map_into_distinguished_fiber¶
map_into_distinguished_fiber:KGBElt x->KGBElt Defined in line number 334.(Auxiliary function)
strong_map_into_distinguished_fiber¶
strong_map_into_distinguished_fiber:KGBElt x->KGBElt Defined in line number 351.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¶
canonical_x_K:KGBElt x->KGBElt Defined in line number 355.Same as previous function.
canonical_x_K¶
canonical_x_K:Param p->KGBElt Defined in line number 358.Previous function with input a parameter p; it is applied to x(p).
u¶
u:KGBElt x->mat Defined in line number 362.Positive coroots in the nilradical of the theta-stable parabolic determined by x.
rho_u_cx¶
rho_u_cx:Parabolic P->ratvec Defined in line number 373.Half sum of positive complex roots (on fundamental Cartan) in the nilradical of P; P must be theta-stable.
rho_u_cx_T¶
rho_u_cx_T:Parabolic P->vec Defined in line number 389.Element of \(X^*\) with same restriction to \((X^*)^{\theta}\) as rho_u_cx(P); P must be theta-stable.
rho_u_ic¶
rho_u_ic:Parabolic P->ratvec Defined in line number 399.Half sum of imaginary compact roots in nilradical of (theta-stable) P.
two_rho_u_cap_k¶
two_rho_u_cap_k:Parabolic P->vec Defined in line number 407.Sum of compact roots (of \(\mathfrak t\) ) in \(\mathfrak u\) for theta-stable parabolic P.
two_rho_u_cap_s¶
two_rho_u_cap_s:Parabolic P->vec Defined in line number 411.Sum of non-compact roots in \(\mathfrak u\) (for theta-stable parabolic).
rho_u_cap_k¶
rho_u_cap_k:Parabolic P->ratvec Defined in line number 416.Half sum of compact roots in \(\mathfrak u\) (for theta-stable parabolic).
rho_u_cap_s¶
rho_u_cap_s:Parabolic P->ratvec Defined in line number 419.Half sum of non-compact roots in \(\mathfrak u\) (for theta-stable parabolic).
dim_u¶
dim_u:Parabolic P->int Defined in line number 422.Dimension of \(\mathfrak u\) (nilrad of theta-stable parabolic).
dim_u¶
dim_u:KGBElt x->int Defined in line number 425.Dimension of the nilradical of the theta-stable parabolic determined by KGB elt x.
dim_u_cap_k¶
dim_u_cap_k:Parabolic (,x):P->int Defined in line number 431.Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic.
dim_u_cap_k¶
dim_u_cap_k:KGBElt x->int Defined in line number 442.Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic determined by x.
dim_u_cap_k¶
dim_u_cap_k:ratvec lambda,KGBElt x->int Defined in line number 446.Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic determined by weight lambda.
dim_u_cap_p¶
dim_u_cap_p:Parabolic (,x):P->int Defined in line number 451.Dimension of \(\mathfrak u\cap\mathfrak p\) for theta-stable parabolic.
dim_u_cap_p¶
dim_u_cap_p:KGBElt x->int Defined in line number 462.Dimension of \(\mathfrak u \cap\mathfrak p\) for theta-stable parabolic associated to x.
dim_u_cap_p¶
dim_u_cap_p:ratvec lambda,KGBElt x->int Defined in line number 466.Dimension of \(\mathfrak u\cap\mathfrak p\) for theta-stable parabolic determined by weight lambda.
dim_u_cap_k_2¶
dim_u_cap_k_2:Parabolic P,ratvec H->int Defined in line number 471.(Auxiliary function)
dim_u_cap_k_ge2¶
dim_u_cap_k_ge2:Parabolic P,ratvec H->int Defined in line number 482.(Auxiliary function)
dim_u_cap_p_ge2¶
dim_u_cap_p_ge2:Parabolic P,ratvec H->int Defined in line number 493.(Auxiliary function)
dim_u_cap_k_1¶
dim_u_cap_k_1:Parabolic P,ratvec H->int Defined in line number 504.(Auxiliary function)
make_dominant¶
make_dominant:KGBElt x_in,ratvec lambda_in, ratvec lambda_q_in->(KGBElt,ratvec,ratvec) Defined in line number 537.Conjugate the triple (x,lambda, lambda_q) to make lambda_q weakly dominant (auxiliary function).
Aq_reducible¶
Aq_reducible:KGBElt x_in,ratvec lambda_in, ratvec lambda_q->ParamPol Defined in line number 544.A_q(lambda) module; \(\mathfrak q\) 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¶
Aq:KGBElt x_in,ratvec lambda_in, ratvec lambda_q->Param Defined in line number 566.A_q(lambda) module defined as above, but as a parameter, assuming it is irreducible.
Aq¶
Aq:KGBElt x,ratvec lambda_in->Param Defined in line number 574.If not provided, assume lambda_q=lambda_in in the definition of A_q.
Aq¶
Aq:RealForm G,ratvec lambda_in, ratvec lambda_q->Param Defined in line number 578.A_q(lambda), specify G, not x, to use x=KGB(G,0).
Aq¶
Aq:RealForm G,ratvec lambda_in->Param Defined in line number 582.A_q(lambda), specify G, not x, and use lambda_q=lambda_in.
is_one_dimensional¶
is_one_dimensional:Param p->bool Defined in line number 589.Decide whether a parameter defines a one-dimensional representation.
is_unitary_character¶
is_unitary_character:Param p->bool Defined in line number 593.Decide whether a parameter defines a unitary one-dimensional character.
is_good¶
is_good:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool Defined in line number 600.Decide whether A_q(lambda) is good.
is_weakly_good¶
is_weakly_good:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool Defined in line number 605.Decide whether A_q(lambda) is weakly good.
is_fair¶
is_fair:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool Defined in line number 610.Decide whether A_q(lambda) is fair.
is_weakly_fair¶
is_weakly_fair:KGBElt x_in,ratvec lambda_in,ratvec lambda_q_in->bool Defined in line number 615.Decide whether A_q(lambda) is weakly fair.
goodness¶
goodness:KGBElt x,ratvec lambda_in,ratvec lambda_q->string Defined in line number 621.Determine the “goodness” of an Aq(lambda); returns “good”, “weakly good”, “fair”, “weakly fair”, or “none”.
is_good¶
is_good:Param p_L,RealForm G->bool Defined in line number 637.Decide whether a parameter for L is in the good range for G; this only makes sense if L is the Levi of a (standard) theta-stable parabolic.
is_weakly_good¶
is_weakly_good:Param p_L,RealForm G->bool Defined in line number 651.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¶
is_fair:Param p_L,RealForm G->bool Defined in line number 662.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, and is only defined if p_L is one_dimensional.
is_weakly_fair¶
is_weakly_fair:Param p_L,RealForm G->bool Defined in line number 678.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¶
goodness:Param p_L,RealForm G->string Defined in line number 690.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¶
Aq_packet:RealForm G,ComplexParabolic P->[Param] Defined in line number 706.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¶
Aq_packet:RealForm G,[int] S->[Param]:Aq_packet(G,ComplexParabolic Defined in line number 715.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¶
Aq_packet:RealForm G,[*] S->[Param]:Aq_packet(G,[int] Defined in line number 717.Aq_zeros¶
Aq_zeros:RealForm G->[Param] Defined in line number 721.List all good Aq(0) (inf. char. rho) of G; this is more or less blocku (there may be duplications).
theta_stable_parabolics_max¶
theta_stable_parabolics_max:KGBElt x->[Parabolic] Defined in line number 728.Given a KGB element x, list all theta-stable parabolics in G with maximal element x.
theta_stable_parabolics_with¶
theta_stable_parabolics_with:KGBElt x->[Parabolic] Defined in line number 736.Given a KGB element x, list all theta-stable parabolics in G determined by x.
theta_stable_parabolics_with¶
theta_stable_parabolics_with:[Parabolic] tsp,KGBElt x->[Parabolic] Defined in line number 743.Same as previous function, but takes the output of theta_stable_parabolics(G) as additional input for efficiency.
is_theta_x¶
is_theta_x:KGBElt x->bool Defined in line number 750.Decide whether there is a theta-stable parabolic determined by x.
is_good_range_induced_from¶
is_good_range_induced_from:Param p->[Param] Defined in line number 754.List of parameters p_L in the (weakly) good range for G so that p is theta-induced from p_L; may be more than one.
reduce_good_range¶
reduce_good_range:Param p->(Parabolic,Param) Defined in line number 776.Find the parabolic P and parameter p_L so that p is cohomologically induced, in the (weakly) good range, from p_L, with L minimal (may be G).
is_good_Aq¶
is_good_Aq:Param p->bool Defined in line number 797.Determine whether p is a (weakly) good unitary Aq(lambda).
is_proper_Aq¶
is_proper_Aq:Param p->bool Defined in line number 802.Determine whether p is a proper (weakly) good unitary Aq(lambda).
all_real_induced_one_dimensional¶
all_real_induced_one_dimensional:RealForm G->[Param] Defined in line number 807.