Principal Series and Discrete Series revisited =============================================== Let us review some basic examples :: atlas> set G=SL(2,R) Variable G: RealForm atlas> p=trivial(G) atlas> set p=trivial(G) Variable p: Param atlas> p Value: final parameter(x=2,lambda=[1]/1,nu=[1]/1) atlas> set x=x(p) Variable x: KGBElt atlas> involution (x) Value: | -1 | So the Cartan involution acts by ``-1`` which means the Cartan subgroup is split. This is a minimal spherical principal series with infinitesimal character ``rho``. atlas> infinitesimal_character (p) Value: [ 1 ]/1 atlas> rho(G) Value: [ 1 ]/1 atlas> This is the minimal principal series containing the trivial representation as unique irreducible quotient. On the other end we also talked about the discrete series :: atlas> whattype discrete_series ? Overloaded instances of 'discrete_series' (KGBElt,ratvec)->Param (RealForm,ratvec)->Param atlas> set q=discrete_series (KGB (G,0), rho(G)) Variable q: Param atlas> q Value: final parameter(x=0,lambda=[1]/1,nu=[0]/1) atlas> To find more representations we look at the block of the trivial representation to find other representations of this group :: atlas> print_block (p) Parameter defines element 2 of the following block: 0: 0 [i1] 1 (2,*) *(x=0,lam_rho= [0], nu= [0]/1) e 1: 0 [i1] 0 (2,*) *(x=1,lam_rho= [0], nu= [0]/1) e 2: 1 [r1] 2 (0,1) *(x=2,lam_rho= [0], nu= [1]/1) 1^e atlas> Here the trivial representation is #2 and the other two are discrete series :: atlas> set r=discrete_series (KGB(G,1), rho(G)) Variable r: Param atlas> r Value: final parameter(x=1,lambda=[1]/1,nu=[0]/1) atlas> set x_b=KGB(G,0) Variable x_b: KGBElt atlas> hc_parameter (q,x_b) Value: [ 1 ]/1 atlas> atlas> hc_parameter (r,x_b) Value: [ -1 ]/1 atlas> So, the Harish-Chandra parameter of ``q`` is ``1`` and that of ``r`` is ``-1``; the holomorphic and antiholomorphic one respectively. But, recall there is another representation with infinitesimal character ``rho`` which is not in the trivial block :: 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=2,lambda=[1]/1,nu=[1]/1) final parameter(x=2,lambda=[2]/1,nu=[1]/1) final parameter(x=1,lambda=[1]/1,nu=[0]/1) final parameter(x=0,lambda=[1]/1,nu=[0]/1) atlas> And recall that the second representation in this list corresponds to the irreducible non-spherical principal series of :math:`SL(2,R)` :: atlas> print_block (params[1]) Parameter defines element 0 of the following block: 0: 0 [rn] 0 (*,*) *(x=2,lam_rho= [1], nu= [1]/1) 1^e atlas> So, this representation is a singleton block. In any case, other than principal series or discrete series, there is nothing else for this group at fixed infinitesimal character. Now let us look at another group :: atlas> G:=PGL(2,R) Value: disconnected split real group with Lie algebra 'sl(2,R)' atlas> set p=trivial(G) Variable p: Param (overriding previous instance, which had type Param) atlas> print_block (p) Parameter defines element 1 of the following block: 0: 0 [i2] 0 (1,2) *(x=0,lam_rho= [0], nu= [0]/1) e 1: 1 [r2] 2 (0,*) *(x=1,lam_rho= [0], nu= [1]/2) 1^e 2: 1 [r2] 1 (0,*) *(x=1,lam_rho= [1], nu= [1]/2) 1^e In this case we only have one discrete series, namely number ``0``; and the others are minimal principal series :: atlas> set q=discrete_series (KGB(G,0), rho(G)) Variable q: Param (overriding previous instance, which had type Param) atlas> q Value: final parameter(x=0,lambda=[1]/2,nu=[0]/1) atlas> rho(G) Value: [ 1 ]/2 atlas> atlas> hc_parameter(q) Value: [ 1 ]/2 atlas> Note that :math:`\rho=1/2` in this case. So :math:`X^* +\rho \cong \mathbb Z +1/2` Also there are only two KGB elements in this group :: atlas> print_KGB(G) kgbsize: 2 Base grading: [1]. 0: 0 [n] 0 1 (0)#0 e 1: 1 [r] 1 * (0)#1 1^e atlas> So there is only one :math:`KGB` element to use for the compact Cartan subgroup and this means we only have one discrete series. Equivalently, note that the simple reflection :math:`s_\alpha` is in the Weyl group of :math:`K`, which is disconnected in this case. So :math:`s_\alpha` flips the positive and negative :math:`K` types. On the other hand, we have two principal series in this block associated to the ``KGB`` element ``x=1``. They both have infinitesimal character ``rho``. But they differ in the disconnectedness of :math:`G`. Now to know about more representations we look at other blocks :: atlas> block_sizes (G) Value: | 0, 1 | | 1, 3 | atlas> This says that we have three representations for :math:`PGL(2,R)` at infinitesimal character ``rho`` and we have one extra at a different infinitesimal character. More on the ``block_sizes`` command later.