# Trivial Representation of $$SL(2,R)$$¶

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>