The Character Differential

To talk about the differential of a character let us use the example of a complex torus:

atlas> set H=torus(0,1,0)
Identifier H: RealForm (hiding previous one of type string (constant))

If we have a parameter p we can extract the coordinates of the parameter when needed:

atlas> set p=trivial(H)
Identifier p: Param
atlas> p
Value: final parameter (x=0,lambda=[0,0]/1,nu=[0,0]/1)
atlas> x(p)
Value: KGB element #0
atlas> lambda(p)
Value: [ 0, 0 ]/1
atlas> nu(p)
Value: [0, 0 ]/1

And remember that for now, the important piece of information about x is the Cartan involution of this Cartan:

atlas> involution (x)
| 0, 1 |
| 1, 0 |

Now, when we have a parameter p, we can ask for its infinitesimal character. The answer is of course more interesting for a non-trivial character:

atlas> infinitesimal_character (p)
Value: [ 0, 0 ]/1
atlas> set q=parameter(x,[1,0],[2,-2])
Identifier q: Param (hiding previous one of type Param)
atlas> q
Value: final parameter (x=0,lambda=[1,0]/1,nu=[2,-2]/1)
atlas> infinitesimal_character (q)
Value: [  5, -3 ]/2

If we have q=(x, lambda, nu) the differential of this character is the infinitesimal character which equals \({(1+ \theta )\over 2} \lambda +{(1- \theta )\over 2} \nu\). But nu is already averaged so this equals \({(1+ \theta )\over 2}\lambda +\nu\):

atlas> infinitesimal_character (q)
Value: [  5, -3 ]/2
atlas> (1+theta)*lambda(q)/2
Value: [ 1, 1 ]/2
atlas> (1+theta)*lambda(q)/2+nu(q)
Value: [  5, -3 ]/2

It is less information than lambda and nu. This is because \({(1+ \theta )\over 2}\) looses some of it.