Script Reference
L_packet.at
L-packets for real reductive groups in the sense of the local Langlands correspondence.
Mathematical background
An L-packet is a finite set of irreducible representations of a real group \(G(\mathbb{R})\) with the same L-parameter (homomorphism from the Weil group to the L-group \({}^L G\)).
Definitions
| Name | Signature | Description |
|---|---|---|
| block_decompose | list)=[([Param],[Param])]: | |
| is_valid | y,ratvec gamma)=bool:is_integer(square(y)-gamma) | |
| parameter | x,KGBElt_gen y,ratvec gamma)=Param:parameter(x,gamma-y.torus_factor,gamma) | |
| y_gen | p)=KGBElt_gen: let y=(dual(p.inner_class),-^p.x.involution,p.infinitesimal_character-p.lambda) in | |
| simple_imaginary_roots | x)=mat: | |
| simple_imaginary_reflections | x)=[WeylElt]: | |
| fiber | x)=[KGBElt]: | |
| fiber | G,mat theta)=[KGBElt]: | |
| L_packet | G,ratvec gamma, KGBElt_gen y_gen)=[Param]: | |
| L_packet | p)=[Param]: | |
| L_packet_representative | p)=Param: | |
| L_packet_representatives | params)=[Param]: | |
| L_packet_representatives | P)=[Param]: L_packet_representatives(monomials(P)) | |
| L_packet_stable_sum | p)=ParamPol: | |
| L_packets | params)=[[Param]]: | |
| is_stable_std | P)=bool: | |
| is_stable_irr | P)=bool: | |
| is_stable | P)=bool:is_stable_irr(P) | |
| stable_sums_std | list_of_params)=[ParamPol]: | |
| stable_sums_std_matrices | list)={[([Param],mat)]:} | [([Param],mat)]: |
| change_basis | list,[Param] new_list, mat M)=mat: | |
| irreducibles_as_sums_of_standards | list)={[([Param],mat)]:} | [([Param],mat)]: |
| stable_sums_irr_in_basis_of_standards_as_matrices | list)={([Param],mat):} | ([Param],mat): |
| stable_sums_irr_in_basis_of_standards | list)={([Param],mat):} | ([Param],mat): |
| stable_sums_irr | list)={([Param],mat):} | ([Param],mat): |
| stable_sums_irr_mat | list)=([Param],mat): | |
| show | list,mat M)=void: | |
| show_stable_sums_irr | list)=void: | |
| q | G)=int:rat_as_int((dimension(G) - dimension(K_0(G)))/2) | |
| kottwitz_invariant | G)=int:rat_as_int(q(quasisplit_form(G))-q(G)) | |
| kottwitz_sign | G)=int:(-1)^kottwitz_invariant(G) | |
| inner_lift_std | p,RealForm G)=ParamPol: | |
| inner_lift_std | P,RealForm G)=ParamPol: | |
| inner_lift | P,RealForm G)=ParamPol: | |
| inner_lift_std | p,int i)=ParamPol: inner_lift_std(p,p.real_form.real_forms[i]) | |
| inner_lift_std | P,int i)=ParamPol: inner_lift_std(P,P.real_form.real_forms[i]) | |
| inner_lift | p,int i)=ParamPol: inner_lift(p,p.real_form.real_forms[i]) | |
| inner_lift | P,int i)=ParamPol: inner_lift(P,P.real_form.real_forms[i]) |