Translation PrincipleΒΆ

Now let’s change infinitesimal character using the translation principle. Let us start again with the trivial representation:

atlas> set G=SL(2,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> set p=trivial(G)
Variable p: Param (overriding previous instance, which had type Param)
atlas> p
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas> infinitesimal_character(p)
Value: [ 1 ]/1
atlas> is_finite_dimensional(p)
Value: true
atlas> dimension(p)
Value: 1

We need to use the command T

atlas> whattype T ?
Overloaded instances of 'T'
  (Param,ratvec)->Param
  (ParamPol,ratvec)->ParamPol
atlas>

atlas> set q= T(p,[2])
Variable q: Param (overriding previous instance, which had type Param)

atlas> q
Value: final parameter (x=2,lambda=[2]/1,nu=[2]/1)
atlas> p
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas>

This means translate p from nu = 1 to nu=2 by applying the Zuckerman translation principle. Note that you also changed lambda. This is a feature of the translation principle. What representation is this new translated one?

atlas> is_finite_dimensional(q)
Value: true
atlas> dimension(q)
Value: 2
atlas> infinitesimal_character(q)
Value: [ 2 ]/1
atlas>

So, this way we obtain the two dimensional representation with infinitesimal character 2.

The translation principle is a great tool to move around by changing infinitesimal characters without changing the nature of the representation. For example, a reducible will stay reducible.

In contrast, it is interesting to see what happens when we change nu but keep lambda:

atlas> set q=parameter(KGB(G,2), [1], [0])
Variable q: Param (overriding previous instance, which had type Param)
atlas> q
Value: final parameter (x=2,lambda=[1]/1,nu=[0]/1)
atlas> infinitesimal_character(q)
Value: [ 0 ]/1
atlas>

Comparing composition series of these two we have:

atlas> p
Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1)
atlas> show(composition_series(I(p)))
1*J(x=0,lambda=[1/1],nu=[0/1])
1*J(x=1,lambda=[1/1],nu=[0/1])
1*J(x=2,lambda=[1/1],nu=[1/1])
atlas> q
Value: final parameter (x=2,lambda=[1]/1,nu=[0]/1)
atlas> show(composition_series(I(q)))
1*J(x=2,lambda=[1/1],nu=[0/1])
atlas>

So q is an irreducible spherical principal series at 0. In other words, changing nu without changing lambda changes the reducibility feature of the representation.