thetastable.at Function References¶

map_into_distinguished_fiber:KGBElt x->KGBElt Defined in line number 92.

(Auxiliary function)

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->KGBElt Defined in line number 113.

Same as previous function.

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->mat Defined in line number 120.

Positive coroots in the nilradical of the theta-stable parabolic determined by x.

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->vec Defined in line number 147.

Element of \(X^*\) with same restriction to \((X^*)^{\theta}\) as rho_u_cx(P); P must be theta-stable.

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->vec Defined in line number 165.

Sum of compact roots (of \(\mathfrak t\) ) in \(\mathfrak u\) for theta-stable parabolic P.

two_rho_u_cap_s:Parabolic P->vec Defined in line number 169.

Sum of non-compact roots in \(\mathfrak u\) (for theta-stable parabolic).

rho_u_cap_k:Parabolic P->ratvec Defined in line number 174.

Half sum of compact roots in \(\mathfrak u\) (for theta-stable parabolic).

rho_u_cap_s:Parabolic P->ratvec Defined in line number 177.

Half sum of non-compact roots in \(\mathfrak u\) (for theta-stable parabolic).

dim_u:Parabolic P->int Defined in line number 180.

Dimension of \(\mathfrak u\) (nilrad of theta-stable parabolic).

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->int Defined in line number 189.

Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic.

dim_u_cap_k:KGBElt x->int Defined in line number 200.

Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic determined by x.

dim_u_cap_k:ratvec lambda,KGBElt x->int Defined in line number 204.

Dimension of \(\mathfrak u\cap\mathfrak k\) for theta-stable parabolic determined by weight lambda.

dim_u_cap_p:Parabolic (,x):P->int Defined in line number 209.

Dimension of \(\mathfrak u\cap\mathfrak p\) for theta-stable parabolic.

dim_u_cap_p:KGBElt x->int Defined in line number 220.

Dimension of \(\mathfrak u \cap\mathfrak p\) for theta-stable parabolic associated to x.

dim_u_cap_p:ratvec lambda,KGBElt x->int Defined in line number 224.

Dimension of \(\mathfrak u\cap\mathfrak p\) for theta-stable parabolic determined by weight lambda.

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->int Defined in line number 240.

(Auxiliary function)

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->int Defined in line number 262.

(Auxiliary function)

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->ParamPol Defined in line number 302.

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: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->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->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->Param Defined in line number 340.

A_q(lambda), specify G, not x, and use lambda_q=lambda_in.

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->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->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->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->bool Defined in line number 366.

Decide whether A_q(lambda) is weakly fair.

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->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->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->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->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->string Defined in line number 413.

Determine the “goodness” for \(\mathfrak u\) of a parameter for L; returns “good”, “weakly good”, “fair”, “weakly fair”, or “none”.

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->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->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->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->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] 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,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] Defined in line number 496.