K_highest_weights.at Function References


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=:KHighestWeight (x,mu), KHighestWeight (y,tau)->bool Defined in line number 98.

Equality of KHighestWeights; (x,mu)=(x’,mu’) iff x=x’, and mu and mu’ differ by (1-theta)X^* or compact Weyl group action.

LKTs

LKTs:Param p->[K_Type] Defined in line number 124.

List of LKTs (as final tempered limit parameters) of a module given by a parameter.

LKT

LKT:Param p->K_Type Defined in line number 127.

LKT of module given by a parameter, if it is unique (otherwise error message)

is_split_spherical

is_split_spherical:Param p->bool Defined in line number 134.

Test if parameter p for G is G-spherical: G must be rel. split, all roots in tau(p).

is_split_spherical

is_split_spherical:K_Type p->bool Defined in line number 143.

Test if a K-type is G-spherical: must be LKT of G-spherical p.

highest_weight_split_spherical

highest_weight_split_spherical:K_Type p->KHighestWeight Defined in line number 147.

Highest weight of a G-spherical K-type (necessarily unique).

highest_weight_split_spherical

highest_weight_split_spherical:Param p->KHighestWeight Defined in line number 161.

Highest weight of LKT of a G-spherical parameter (is unique).

highest_weight_split_spherical

highest_weight_split_spherical:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 165.

Highest weight of LKT of G-spherical K-type w.r.t. x_K (is unique).

highest_weight_split_spherical

highest_weight_split_spherical:Param p,KGBElt x_K->KHighestWeight Defined in line number 169.

Highest weight of LKT of a G-spherical parameter w.r.t. x_K (is unique).

highest_weight_one

highest_weight_one:K_Type p->KHighestWeight Defined in line number 173.

This function returns just ONE KHighestWeight of a K-type (auxiliary function).

highest_weight_one

highest_weight_one:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 192.

ONE KHighestWeight of a K-type w.r.t. x_K (auxiliary function).

highest_weights

highest_weights:K_Type p->[KHighestWeight] Defined in line number 197.

All highest weights of a K-type; R-group acts on one KHighestWeight of previous function (number of terms is |R(K)/R(K,mu)|).

highest_weights

highest_weights:K_Type p,KGBElt x_K->[KHighestWeight] Defined in line number 201.

All highest weights of a K-type w.r.t. x_K.

highest_weights

highest_weights:Param p->[KHighestWeight] Defined in line number 205.

List of all highest weights of all LKTs of a parameter.

highest_weights

highest_weights:Param p,KGBElt x_K->[KHighestWeight] Defined in line number 211.

List of all highest weights of all LKTs of a parameter w.r.t. x_K.

highest_weight

highest_weight:K_Type p->KHighestWeight Defined in line number 215.

Unique highest weight of a K-type (or error if not unique).

highest_weight

highest_weight:K_Type p,KGBElt x_K->KHighestWeight Defined in line number 221.

Unique highest weight of a K-type w.r.t. x_K (or error if not unique).

highest_weight

highest_weight:Param p->KHighestWeight Defined in line number 227.

Unique highest weight of (unique) LKT of a parameter (or error if not unique).

centralizer

centralizer:KGBElt x,ratvec v->(KGBElt,RootDatum) Defined in line number 243.

(Auxiliary function)

find_nci_root

find_nci_root:KGBElt x,ratvec tau->int Defined in line number 253.

(Auxiliary function)

tworho_K

tworho_K:KGBElt x->ratvec Defined in line number 287.

Sum of the roots of K as an element of \((X^*)^{\delta}\otimes\mathbb Q\) (this may be half-integral); x must be in the distinguished fiber.

project_on_dominant_cone

project_on_dominant_cone:KGBElt x, ratvec mu->(KGBElt,ratvec,ratvec) Defined in line number 296.

Vogan algorithm to project KHighestWeight (x,mu) on dominant cone; returns (x’,mu+2rho_K(x)-rho,tau) with tau dominant and x’ corresponding to the new Weyl chamber.

project_on_dominant_cone

project_on_dominant_cone:KGBElt x, vec mu->(KGBElt,ratvec,ratvec) Defined in line number 340.

Vogan algorithm; previous function in case mu is given as a vec, rather than ratvec.

characters_order_2

characters_order_2:KGBElt x->[vec] Defined in line number 355.

(Auxiliary function)

all_G_spherical_same_differential

all_G_spherical_same_differential:K_Type p->[K_Type] Defined in line number 375.

All G-spherical K-types with same differential as given one.

all_G_spherical_same_differential

all_G_spherical_same_differential:Param p->[K_Type] Defined in line number 392.

All G-spherical K-types with same differential as the LKT of parameter p.

parabolic_LKT

parabolic_LKT:KHighestWeight (x,mu)->Parabolic Defined in line number 404.

Parabolic attached to KHighestWeight by Vogan algorithm.

make_strongly_dominant

make_strongly_dominant:KHighestWeight mu,KGBElt x_Q->KHighestWeight Defined in line number 419.

(Auxiliary function)

K_types

K_types:KHighestWeight mu_in->[K_Type] Defined in line number 427.

All K_types with the same KHighestWeight.

K_type

K_type:KHighestWeight(x,mu)->K_Type Defined in line number 464.

K_type with given KHighestWeight if unique (otherwise error).

K0_highest_weight

K0_highest_weight:KHighestWeight(x,mu)->Param Defined in line number 474.

Parameter for (the RealForm K_0) of the K_0-type with highest weight (the restriction of) KHighestWeight mu.

dimension

dimension:KHighestWeight mu->int Defined in line number 482.

Dimension of the K_#-type with KHighestWeight mu.

dimension

dimension:K_Type p->int Defined in line number 485.

Dimension of a K-type.

H_weight

H_weight:KGBElt x,vec mu_K->KHighestWeight Defined in line number 496.

(Auxiliary function)

fundamental_weights_K_H

fundamental_weights_K_H:KGBElt x->[ratvec] Defined in line number 501.

(Auxiliary function)

K0_param

K0_param:K_Type p,KGBElt x_K->Param Defined in line number 507.

ONE K_0-type in the restriction of a K_type to the identity component K_0 of K (auxiliary function).

K0_param

K0_param:K_Type p->Param Defined in line number 513.

ONE K_0-type in the restriction of a K_type to the identity component K_0 of K (auxiliary function).

K0_params

K0_params:Param p,KGBElt x_K->[Param] Defined in line number 521.

All K_0-types in the restriction of the LKTs of parameter p K_0.

K0_params

K0_params:Param p->[Param] Defined in line number 526.

All K_0-types in the restriction of the LKTs of parameter p to K_0.

K0_param

K0_param:Param p,KGBElt x_K->Param Defined in line number 530.

Unique K_0-type in the restriction of (unique) LKT to K_0 (error if not unique).

K0_param

K0_param:Param p->Param Defined in line number 536.

Unique K_0-type in the restriction of (unique) LKT to K_0 (error if not unique).

fundamental_weight_coordinates

fundamental_weight_coordinates:KHighestWeight (x,mu)->vec Defined in line number 543.

(Auxiliary function)

K_highest_weight_from_fundamental_weights

K_highest_weight_from_fundamental_weights:KGBElt x,vec tau->KHighestWeight Defined in line number 549.

(Auxiliary function)

dimensions

dimensions:Param p, KGBElt x_K->[int] Defined in line number 558.

List of the dimensions of the K_0-types in the restriction of the LKTs of parameter p.

dimensions

dimensions:Param p->[int] Defined in line number 562.

List of the dimensions of the K_0-types in the restriction of the LKTs of parameter p.

dimensions

dimensions:[Param] B->[[int]] Defined in line number 566.

List of the dimensions of the K_0-types in the restriction of the LKTs of a list of parameters.