# parabolics.at Function References¶

## sort_by¶

sort_by:(KGBElt -> int) f->([KGBElt] v) [KGBElt] Defined in line number 61.

Given a list of KGB elements and a function f assigning integers to them, sort the list by weakly increasing value of f.

## KGP_elt¶

KGP_elt:KGPElt pair->KGPElt Defined in line number 75.

## S¶

S:KGPElt(S,)->[int] Defined in line number 78.

The list S of simple roots of a KGP element.

## root_datum¶

root_datum:KGPElt(,x)->RootDatum Defined in line number 81.

The root datum of the RealForm G of a KGP element.

## real_form¶

real_form:KGPElt(,x)->RealForm Defined in line number 84.

The RealForm G of a KGP element.

## complement¶

complement:int n,[int] S->[int] Defined in line number 87.

Complement of subset of simple roots in rank n.

## find_ascent¶

find_ascent:[int] S, KGBElt x->[KGBElt] Defined in line number 91.

An ascent of x by a generator in S, if any exist.

## down_neighbors¶

down_neighbors:[int] S,KGBElt x->[int] Defined in line number 99.

All descents of x by generators in S; there may be duplicates.

## is_maximal_in_partial_order¶

is_maximal_in_partial_order:[int] S,KGBElt x->bool Defined in line number 110.

Decide whether x is maximal in the partial order defined by S.

## maxima_in_partial_order¶

maxima_in_partial_order:RealForm G,[int] S->[KGBElt] Defined in line number 113.

List maximal KGB elements in the partial order defined by S.

## maximal¶

maximal:[int] S, KGBElt x->KGBElt Defined in line number 119.

(Unique) maximal element in equivalence class of x.

## canonical_representative¶

canonical_representative:KGPElt y->KGPElt Defined in line number 124.

The representative of a KGP element with maximal x.

## =¶

=:KGPElt (S,x),KGPElt (T,y)->bool Defined in line number 131.

Equality of KGP elements: (S,x)=(T,y) if these give the same K-orbit of parabolics.

## equivalence_class_of¶

equivalence_class_of:KGPElt(S,x)->[KGBElt] Defined in line number 135.

The equivalence class of a KGB element in partial order defined by S.

## rec_fun x_min¶

rec_fun x_min:KGPElt P->KGBElt Defined in line number 149.

A minimal KGB element from an equivalence class defined by S (unlike x_max, it is not unique).

## KGP¶

KGP:RealForm G,[int] S->[KGPElt] Defined in line number 155.

The set of KGP elements associated to a RealForm and a set of simple roots S; KGP(G,S) is in bijection with $$K\backslash G/P_S$$ .

## KGP_numbers¶

KGP_numbers:RealForm G,[int] S->[int] Defined in line number 159.

Just the index numbers (maximal x) of KGP(G,S).

## is_open¶

is_open:KGPElt y->bool Defined in line number 163.

Test whether y in $$K\backslash G/P_S$$ is open: <=> last element of y is last element of KGB.

## is_closed¶

is_closed:KGPElt P->bool Defined in line number 166.

Test whether y in $$K\backslash G/P_S$$ is closed: <=> length(first element)=0.

## KGP_elt¶

KGP_elt:ratvec lambda,KGBElt x->KGPElt Defined in line number 169.

Parabolic determined by (the stabilizer in W of) a weight lambda.

## complex_parabolic¶

complex_parabolic:Parabolic(S,x)->ComplexParabolic Defined in line number 179.

The complex parabolic underlying P=(S,x).

## complex_Levi¶

complex_Levi:RootDatum rd, (int->bool) select->RootDatum Defined in line number 182.

Auxiliary function

## is_Levi_theta_stable¶

is_Levi_theta_stable:Parabolic (S,x)->bool Defined in line number 192.

Test if a complex Levi defined by a set of simple roots S is $$\theta_x$$ -stable; algorithm: H=sum of fundamental coweights with index not in S, test whether $$<\theta_x(\alpha),H>=0$$ for all $$\alpha$$ in S.

## Levi¶

Levi:Parabolic(S,x):P->RealForm Defined in line number 204.

Make a real Levi factor from P=(S,x); the complex Levi of S must be theta-stable.

## is_parabolic_theta_stable¶

is_parabolic_theta_stable:Parabolic (S,x):P->bool Defined in line number 213.

Test if parabolic P=(S,x) is theta-stable: <=> the complex Levi factor L is theta-stable, P is closed, and for alpha simple, not in S => alpha is imaginary or C+ wrt maximal(P).

## is_parabolic_real¶

is_parabolic_real:Parabolic (S,x):P->bool Defined in line number 224.

Test if parabolic P=(S,x) is real: <=> L is theta-stable, P is open, and for alpha simple, not in S => alpha is real or C- wrt a maximal(P).

## rho_u¶

rho_u:ComplexParabolic P->ratvec Defined in line number 245.

Half sum of positive roots not in the Levi (L must be theta-stable).

## rho_u¶

rho_u:Parabolic P->ratvec Defined in line number 248.

Half sum of positive roots not in the Levi (L must be theta-stable).

## rho_l¶

rho_l:Parabolic P->ratvec Defined in line number 251.

Half sum of positive roots in the Levi (L must be theta-stable).

nilrad:Parabolic P->mat Defined in line number 254.

Positive coroots in the nilradical u of P (L must be theta-stable).

nilrad_roots:Parabolic P->mat Defined in line number 259.

Positive roots in the nilradical u of P (L must be theta-stable).

## zero_simple_coroots¶

zero_simple_coroots:RootDatum rd, vec lambda->[int] Defined in line number 272.

Simple coroots on which weight lambda (in $$\mathfrak h^*$$ ) is zero.

## parabolic¶

parabolic:ratvec lambda,KGBElt x->Parabolic Defined in line number 279.

Parabolic defined by weight lambda; message whether parabolic is real or theta-stable.

## parabolic_mute¶

parabolic_mute:ratvec lambda,KGBElt x->Parabolic Defined in line number 290.

Parabolic defined by weight lambda; NO message whether parabolic is real or theta-stable.

## theta_stable_parabolic¶

theta_stable_parabolic:ratvec lambda,KGBElt x->Parabolic Defined in line number 297.

Theta-stable parabolic defined by weight lambda.

## real_parabolic¶

real_parabolic:ratvec lambda,KGBElt x->Parabolic Defined in line number 301.

Real parabolic defined by weight lambda.

## Levi¶

Levi:ratvec lambda,KGBElt x->RealForm Defined in line number 305.

Levi factor of parabolic defined by weight lambda.

## theta_stable_Levi¶

theta_stable_Levi:ratvec lambda, KGBElt x->RealForm Defined in line number 308.

Levi factor of theta-stable parabolic defined by weight lambda.

## real_Levi¶

real_Levi:ratvec lambda, KGBElt x->RealForm Defined in line number 312.

Levi factor of real parabolic defined by weight lambda.

nilrad:ratvec lambda,KGBElt x->mat Defined in line number 316.

Positive coroots in nilradical of P defined by weight lambda (if L theta-stable).

nilrad_roots:ratvec lambda,KGBElt x->mat Defined in line number 319.

Positive roots in nilradical of P defined by weight lambda (if L theta-stable).

## rho_u¶

rho_u:ratvec lambda,KGBElt x->ratvec Defined in line number 324.

Half sum of positive roots in nilradical of P defined by weight lambda (if L theta-stable).

## zero_simple_roots¶

zero_simple_roots:RootDatum rd, vec cowt->[int] Defined in line number 327.

Simple roots which are zero on coweight H (in $$\mathfrak h$$ ).

## parabolic_alt¶

parabolic_alt:ratvec H,KGBElt x->Parabolic Defined in line number 334.

Parabolic defined by coweight H; message whether parabolic is real or theta-stable.

## Levi_alt¶

Levi_alt:ratvec H,KGBElt x->RealForm Defined in line number 344.

Levi factor of parabolic defined by coweight H.

nilrad_alt:ratvec H,KGBElt x->mat Defined in line number 347.

Positive coroots in nilradical of P defined by coweight H (if L theta-stable).

nilrad_roots_alt:ratvec H,KGBElt x->mat Defined in line number 350.

Positive roots in nilradical of P defined by coweight H (if L theta-stable).

## rho_u_alt¶

rho_u_alt:ratvec H,KGBElt x->ratvec Defined in line number 354.

Half sum of roots in nilradical of P defined by coweight H (if L theta-stable).

## rho_Levi_alt¶

rho_Levi_alt:ratvec H,KGBElt x->ratvec Defined in line number 357.

$$\rho(L)$$ for Levi of P defined by coweight H (if L theta-stable).

## real_parabolic¶

real_parabolic:KGBElt x->Parabolic Defined in line number 365.

Real parabolic defined by x has Levi factor M=centralizer(A), $$\mathfrak u$$ =positive roots not in M; for M to be stable: x must have no C+ roots.

## real_Levi¶

real_Levi:KGBElt x->RealForm Defined in line number 370.

Levi factor of real parabolic defined by x (must have no C+ roots).

## theta_stable_parabolic¶

theta_stable_parabolic:KGBElt x->Parabolic Defined in line number 377.

Theta-stable parabolic defined by x has Levi factor L=centralizer(T), $$\mathfrak u$$ =positive roots not in L; for this to be stable: no C- roots.

## theta_stable_Levi¶

theta_stable_Levi:KGBElt x->RealForm Defined in line number 382.

Levi factor of theta-stable parabolic defined by x (must have no C- roots).

## is_standard_Levi¶

is_standard_Levi:RealForm L,RealForm G->bool Defined in line number 386.

Check whether a Levi subgroup L is standard in G (simple roots of L are simple for G).

## KGP¶

KGP:RealForm G,ComplexParabolic (rd,S)->[KGPElt] Defined in line number 395.

List of K-conjugacy classes of given ComplexParabolic (as KGP elts).

## parabolics¶

parabolics:RealForm G,ComplexParabolic (rd,S)->[Parabolic] Defined in line number 399.

List K-conjugacy classes of given ComplexParabolic (as Parabolics).

## theta_stable_parabolics¶

theta_stable_parabolics:RealForm G,ComplexParabolic P->[Parabolic] Defined in line number 403.

List K-conjugacy classes of given ComplexParabolic that are theta-stable.

## theta_stable_parabolics¶

theta_stable_parabolics:RealForm G->[Parabolic] Defined in line number 409.

List all theta-stable parabolics for G.

## theta_stable_parabolics_type¶

theta_stable_parabolics_type:RealForm G,[int] P->[Parabolic] Defined in line number 416.

List all theta-stable parabolics of G, of type S.

## all_rel_split_theta_stable_parabolics¶

all_rel_split_theta_stable_parabolics:RealForm G->[Parabolic] Defined in line number 422.

List all theta-stable parabolics of G with relatively split L.

## support¶

support:KGBElt x->[int] Defined in line number 441.

The smallest list of simple roots such that descents lead to the distinguished fiber.

## support_alt¶

support_alt:KGBElt x->[int] Defined in line number 450.

Auxiliary function.

## KGPElt¶

([int], KGBElt) Defined in line number 54.

Data type for a K_orbit on G/P_S, equivalently a K-conjugacy class of parabolics of type S.

## Parabolic¶

([int], KGBElt) Defined in line number 57.

Data type for a K_orbit on G/P_S (synonym for KGPElt).

## ComplexParabolic¶

(RootDatum,[int]) Defined in line number 176.

Data type for a complex parabolic subrgoup