# 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.