Trivial Representation of \(SL(2,R)\)ΒΆ

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>