Let us consider again the case of \(SL(2,R)\) and the trivial representation.:
atlas> set G=SL(2,R)
Identifier G: RealForm
atlas> G
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas> p:=trivial(G)
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas> x:=x(p)
Value: KGB element #2
atlas> theta:=involution(x)
Value:
| -1 |
atlas>
So the parameter for the trivial representation contains information
of the Cartan subgroup and its cartan involution, \(\theta\),
encoded in the \(K\backslash G/B\) element x. In this case
\(\theta=-1\). This means it is the split Cartan subgroup, which
is isomorphic to \({\mathbb R }^x\)
We also have encoded information about the character which, as we saw in the
section on characters of real tori, is given by lambda and
nu. Here nu=1 is the differential of the character, and
lambda=1 gives the character on the component group \({\mathbb
Z}/(1-\theta){\mathbb Z}=\mathbb Z/2{\mathbb Z}\), of the torus:
atlas> (1+theta)*lambda(p)/2
Value: [ 0 ]/1
atlas> (1-theta)*nu(p)/2
Value: [ 1 ]/1
atlas>