More parameter commands ======================== Principal series commands -------------------------- There is another command which we will use here to look at more examples of minimal principal series. Namely, in addition to the command ``all_minimal_principal_series``, the command ``minimal_principal_series`` helps us identify a particular representation in the series. Let us compare their use with some examples:: atlas> set G=Sp(4,R) Variable G: RealForm atlas> whattype minimal_principal_series ? Overloaded instances of 'minimal_principal_series' (RealForm,ratvec,ratvec)->Param RealForm->Param We will use the first syntax above:: atlas> minimal_principal_series(G,rho(G),rho(G)) Value: final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) atlas> atlas> minimal_principal_series(G,rho(G),[0,0]) Value: final parameter (x=10,lambda=[2,1]/1,nu=[0,0]/1) So we get the single trivial or the representation with ``nu=0``. Now, recall that for the first command, we need to provide a real form and a rational vector:: atlas> whattype all_minimal_principal_series ? Overloaded instances of 'all_minimal_principal_series' (RealForm,ratvec)->[Param] atlas> atlas> set ps= all_minimal_principal_series (G,rho(G)) Variable ps: [Param] atlas> atlas> void: for p in ps do prints(p) od final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1) final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1) final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1) atlas> So, in this case we obtain again the four principal series of :math:`Sp(4,R)` at infinitesimal character ``rho``. The ``nus`` all equal ``rho`` and the ``lambdas`` are all the possible lambdas in :math:`X^*/2X^*`. Note that the group does not have to be semisimple:: atlas> G:=GL(2,R) Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' atlas> set ps= all_minimal_principal_series (G,rho(G)) Variable ps: [Param] (overriding previous instance, which had type [Param]) atlas> void: for p in ps do prints(p) od final parameter(x=1,lambda=[1,-1]/2,nu=[1,-1]/2) final parameter(x=1,lambda=[3,-1]/2,nu=[1,-1]/2) final parameter(x=1,lambda=[1,1]/2,nu=[1,-1]/2) final parameter(x=1,lambda=[3,1]/2,nu=[1,-1]/2) atlas> WARNING: This command does not work for non-split groups:: atlas> G:=U(2,2) Value: connected quasisplit real group with Lie algebra 'su(2,2).u(1)' atlas> set ps= all_minimal_principal_series (G,rho(G)) group is not split (in call at atlas-scripts/basic.at:8:57-71 of error@string, built-in) [b=false, message="group is not split"] (in call at atlas-scripts/all_parameters.at:109:4-44 of assert@(bool,string), defined at atlas-scripts/basic.at:8:4-74) [G=connected quasisplit real group with Lie algebra 'su(2,2).u(1)', gamma= [ 3, 1, -1, -3 ]/2] (in call at :5:7-45 of all_minimal_principal_series@(RealForm, ratvec), defined at atlas-scripts/all_parameters.at:108:4--110:63) Command 'set ps' interrupted, nothing defined. atlas> ``all_parameters_gamma`` ------------------------ For this group we need to use the command that lists all representations with a given parameter for :math:`G` :: atlas> G:=U(2,2) Value: connected quasisplit real group with Lie algebra 'su(2,2).u(1)' atlas> set params=all_parameters_gamma (G,rho(G)) Variable params: [Param] (overriding previous instance, which had type [Param]) atlas> #params Value: 21 atlas> atlas> void: for p in params do prints(p) od final parameter(x=20,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2) final parameter(x=19,lambda=[3,1,-1,-3]/2,nu=[3,0,0,-3]/2) final parameter(x=18,lambda=[3,1,-1,-3]/2,nu=[3,0,0,-3]/2) final parameter(x=17,lambda=[3,1,-1,-3]/2,nu=[1,1,-1,-1]/1) final parameter(x=16,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1) final parameter(x=15,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1) final parameter(x=14,lambda=[3,1,-1,-3]/2,nu=[0,1,0,-1]/1) final parameter(x=13,lambda=[3,1,-1,-3]/2,nu=[0,1,0,-1]/1) final parameter(x=12,lambda=[3,1,-1,-3]/2,nu=[1,-1,1,-1]/2) final parameter(x=11,lambda=[3,1,-1,-3]/2,nu=[1,-1,0,0]/2) final parameter(x=10,lambda=[3,1,-1,-3]/2,nu=[1,-1,0,0]/2) final parameter(x=9,lambda=[3,1,-1,-3]/2,nu=[0,1,-1,0]/2) final parameter(x=8,lambda=[3,1,-1,-3]/2,nu=[0,1,-1,0]/2) final parameter(x=7,lambda=[3,1,-1,-3]/2,nu=[0,0,1,-1]/2) final parameter(x=6,lambda=[3,1,-1,-3]/2,nu=[0,0,1,-1]/2) final parameter(x=5,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) final parameter(x=4,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) final parameter(x=3,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) final parameter(x=2,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) final parameter(x=1,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) final parameter(x=0,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) atlas> Recall that all Cartan subgroups of :math:`U(2,2)` are connected. And we can find the information on the Cartan subgroup associated to each parameter as follows :: atlas> p:=trivial(G) Value: final parameter(x=20,lambda=[3,1,-1,-3]/2,nu=[3,1,-1,-3]/2) atlas> atlas> H:=Cartan_class(p) Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms atlas> atlas> print_Cartan_info (H) compact: 0, complex: 2, split: 0 canonical twisted involution: 2,1,3,2 twisted involution orbit size: 3; fiber size: 1; strong inv: 3 imaginary root system: empty real root system: A1.A1 complex factor: A1 atlas> This is the most split Cartan subgroup in :math:`U(2,2)`. It is just two copies of :math:`{\mathbb C}^x`. So it is connected. In fact this group has three minimal principal series (with ``x=17`` and ``x=12``) not comming from the disconnectedness of the Cartan subgroup but from the Weyl group. We will address this later. ``all_parameters`` ------------------- This command helps us find representations with the same differential :: atlas> G:=Sp(4,R) Value: connected split real group with Lie algebra 'sp(4,R)' atlas> set params=all_parameters_gamma (G,rho(G)) Variable params: [Param] (overriding previous instance, which had type [Param]) atlas> void: for p in params do prints(p) od final parameter(x=10,lambda=[2,1]/1,nu=[2,1]/1) final parameter(x=10,lambda=[3,1]/1,nu=[2,1]/1) final parameter(x=10,lambda=[2,2]/1,nu=[2,1]/1) final parameter(x=10,lambda=[3,2]/1,nu=[2,1]/1) final parameter(x=9,lambda=[2,1]/1,nu=[3,3]/2) final parameter(x=8,lambda=[2,1]/1,nu=[2,0]/1) final parameter(x=8,lambda=[3,1]/1,nu=[2,0]/1) final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1) final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1) final parameter(x=6,lambda=[2,1]/1,nu=[0,1]/1) final parameter(x=6,lambda=[2,2]/1,nu=[0,1]/1) final parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1) final parameter(x=5,lambda=[2,2]/1,nu=[0,1]/1) final parameter(x=4,lambda=[2,1]/1,nu=[1,-1]/2) final parameter(x=3,lambda=[2,1]/1,nu=[0,0]/1) final parameter(x=2,lambda=[2,1]/1,nu=[0,0]/1) final parameter(x=1,lambda=[2,1]/1,nu=[0,0]/1) final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1) atlas> atlas> p:=params[8] Value: final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1) atlas> set others=all_parameters (p) Variable others: [Param] (overriding previous instance, which had type [Param]) atlas> void: for p in others do prints(p) od final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1) final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1) atlas> void: for q in others do prints(q) od final parameter(x=7,lambda=[3,1]/1,nu=[2,0]/1) final parameter(x=7,lambda=[2,1]/1,nu=[2,0]/1) atlas> This Cartan subgroup has two connected components. So if you hand in a parameter for this subgroup, the total number of parameters with the same differential is two and this command gives the list of all of them.