Parameters for Real Tori

To study characters of Tori we first need to know how atlas understands what a torus is. In order to specify which torus we want we use the command torus. There are two ways of using this command, which we can check as follows:

atlas> whattype torus ?
Overloaded instances of 'torus'
  (int,int,int)->RealForm
  CartanClass->RealForm
atlas>

We are interested in the first option. So, we can determine a particular torus by assigning three integers:

atlas> H:=torus(1,0,0)
Value: compact connected quasisplit real group with Lie algebra 'u(1)'
atlas>
atlas> H:=torus(0,1,0)
Value: connected quasisplit real group with Lie algebra 'gl(1,C)'
atlas>
atlas> H:=torus(0,0,1)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas>
atlas> H:=torus(1,2,3)
Value: disconnected quasisplit real group with Lie algebra 'u(1).gl(1,C).gl(1,C).gl(1,R).gl(1,R).gl(1,R)'
atlas>

In other words, the command torus(a,b,c) specifies a torus with a \(S^1\) factors, b \({\mathbb C}^{\times }\) factors and c \({\mathbb R}^{\times }\) factors.

The characters of \(S^1\)

Starting with the circle, let us discuss its representations. From the theory we know that they are parametrized by integers. So, starting from the trivial representation we can see how the software parametrizes them:

atlas>  H:=torus(1,0,0)
Value: compact connected quasisplit real group with Lie algebra 'u(1)'
atlas> set p=trivial(H)
Identifier p: Param
atlas> p
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas>

This is the parameter for the trivial representation. The first element x is a \(K\backslash G/B\) element. But it is not important for now. Let us see what involution it corresponds to:

atlas> set x=x(p)
Identifier x: KGBElt
atlas> set theta=involution(x)
Identifier theta: mat
atlas> theta
Value:
| 1 |

atlas>

In this case it is the identity involution. That is, the torus has a Cartan involution whose information is encoded in this element x and in the case of a compact torus it is the identity.

Now to understand the rest of the parameters, suppose that \(H\) is a complex torus with Cartan involution \(\theta\), and defined over \(\mathbb R\). Let \(H(\mathbb R )\) be the corresponding real group. Denote by \(X^* (H)\) the characters of \(H(\mathbb R )\). Then these characters are parametrized by triples \((\theta, \lambda ,\nu)\). Here \(\nu\) is the differential of the character and lambda is the restriction of the character to \(H^{\theta}\).

The real points are not necessarily connected. So we need to specify not only the differential but also information of the disconnected part which is encoded in the restriction to \(H^{\theta}\).

The basic fact is that the characters of \(H^{\theta}\) are parametrized by \(X^*/(1-\theta)X^*\).

So the characters of \(H(\mathbb R )\) are parametrized by triples \((\theta, \lambda, \nu)\) where

\[\lambda \in X^* /(1-\theta )X^*\]

and

\[\nu \in [X^*\otimes \mathbb Q]^{-\theta}.\]

So lambda is a character on \(H^\theta\) and nu gives the character on the Lie algebra of the split Cartan subgroup \(A=H^{-\theta}\).

In the case of the circle, lambda is in \(\mathbb Z/(1-\theta ){\mathbb Z}=\mathbb Z\), since \(\theta =1\); and nu is in \({\mathbb Q}^{-1}=0\).

So the characters for the compact torus are given by the parameters (x=0, lambda, nu=0) and there is one for each lambda in \(\mathbb Z\) correspondintg to the character \(e^{i\lambda t}\).

In the case of the trivial character the parameter is:

atlas> p
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas>

And for the character \(e^{i 3} t\) of \(S^1\) we have:

atlas> q:=parameter (x,[3],[0])
Value: final parameter (x=0,lambda=[3]/1,nu=[0]/1)
atlas>

Note that nu is fixed by \(-\theta\). So, given any nu it will be replaced by \(\frac{1-\theta}{2} \nu \in (X^*_{\mathbb Q})^{-\theta}\):

atlas> q:=parameter (x,[3],[2])
Value: final parameter (x=0,lambda=[3]/1,nu=[0]/1)
atlas>

So, the above parameters are equivalent modulo the above equivalence relation and parametrize the same character. For example, we can ask atlas if

atlas> parameter (x,[3],[3])= parameter (x,[3],[0])
Value: true
atlas>

The characters of \({\mathbb R}^{\times}\)

Now let us take the most split one-dimensional torus:

atlas> H:=torus(0,0,1)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> p:=trivial (H)
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas> set x=x(p)
Identifier x: KGBElt (hiding previous one of type KGBElt)
atlas> theta:=involution(x)
Value:
| -1 |

Now our parameters (x, lambda, nu) satisfy lambda is in \({\mathbb Z}/(1-\theta){\mathbb Z}=\mathbb Z/2{\mathbb Z}\), and nu is fixed by \(-\theta=1\). So, nu is in \(\mathbb Q\).

Note that the characters of \({\mathbb R}^{\times}\) are parametrized by the complex numbers nu. However, the software only works with rational parameters. So we have to do some extra work in general, depending on the information that we want. The idea is that some problems can be reduced to the case of rational parameters.

Let us do some examples. For the trivial representation, namely the parameter correspondintg to the trivial character on the component group and the character \(\nu :x \rightarrow |x|^0\) we have:

atlas> p
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas>

And for the representation with \(\nu :x \rightarrow |x|^{4/3}\)

atlas> p:=parameter (x,[0],[4/3])
Value: final parameter (x=0,lambda=[0]/1,nu=[4]/3)
atlas>

Now suppose we want a representation with non trivial character on the component group \({\mathbb Z}/2{\mathbb Z}\). For example, the sign representation is given by:

atlas> q:=parameter (x,[1],[0])
Value: final parameter (x=0,lambda=[1]/1,nu=[0]/1)
atlas>

atlas> p:=trivial(H)
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas>

Which differs from the trivial by the non trivial character on \({\mathbb Z}/2{\mathbb Z}\). And note what happens when we change that character to 2:

atlas> q:=parameter (x,[2],[0])
Value: final parameter (x=0,lambda=[0]/1,nu=[0]/1)
atlas> p=q
Value: true
atlas>

Which is correct since \(2=0(mod2)\). So, sometimes the software will replace the parameters you are using for something equivalent.

Characters of \({\mathbb C}^{\times}\).

Now let us look at \({\mathbb C}^{\times} \cong GL(1, \mathbb C)\) and the trivial representation:

atlas> H:=torus(0,1,0)
Value: connected quasisplit real group with Lie algebra 'gl(1,C)'
atlas> set p=trivial(H)
Identifier p: Param
atlas> p
Value: final parameter (x=0,lambda=[0,0]/1,nu=[0,0]/1)

Now we have two coordinates for each parameter because we have a rank-2 real group, locally isomorphic to \(S^1 \times {\mathbb R}^{\times}\). Let’s see what the Cartan involution is for this torus:

atlas> set x=x(p)
Identifier x: KGBElt
atlas> set theta=involution (x)
Identifier theta: mat
atlas> theta
Value:
| 0, 1 |
| 1, 0 |

atlas>

So the Cartan involution of the complex torus switches the two coordinates. For example if lambda = [0,0] and nu= [2,4], we have:

atlas> set q=parameter (x,[0,0],[2,4])
Identifier q: Param (hiding previous one of type vec (constant))
atlas> q
Value: final parameter (x=0,lambda=[0,0]/1,nu=[-1,1]/1)
atlas>

Here the software leaves lambda as [0,0] and it changes nu to [-1,1]. Which makes sense since``nu`` is fixed by \(-\theta\) so it changed nu to \(\frac{1-\theta}{2} \nu\):

atlas> (1-theta)*[2,4]/2
Value: [ -1,  1 ]/1
atlas>

So, in fact for this group the nu will always look like [x,-x]:

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

On the other hand, we can change lambda:

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

As we would expect since these two representations are equivalent modulo \(1-\theta\)

So, The representations of \({\mathbb C}^{\times}\) are given by \({\mathbb Z}^2 /(1-\theta) {\mathbb Z}^2\) and \(\mathbb Q\)