Tutorial

Representations with Zero Infinitesimal Character

Let \(G=Sp(4, \mathbb R)\). We can also compute all the representations with infinitesimal character zero

atlas> G:Sp(4,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> set parameters=all_parameters_gamma(G,[0,0])
Variable parameters: [Param] (overriding previous instance, which had type [Param])
atlas> #parameters
Value: 5
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[0,0]/1,nu=[0,0]/1)
final parameter (x=1,lambda=[0,0]/1,nu=[0,0]/1)
final parameter (x=5,lambda=[0,1]/1,nu=[0,0]/1)
final parameter (x=6,lambda=[0,1]/1,nu=[0,0]/1)
final parameter (x=10,lambda=[2,1]/1,nu=[0,0]/1)
atlas>

Note that nu is always 0. The last row is the spherical principal series at infinitesimal character 0 and lambda is essentially rho. But this is only telling you what the character is on the \({\mathbb Z}_2\) factors. On the other end the first two are the limits of discrete series and we have two intermediate ones. So we only have one principal series at zero infinitesimal character.