Cartan Subgroups

When we give atlas a specific Lie group \(G\), the software knows several characteristics of the group. And depending on what you ask about \(G\), atlas will provide more information:

atlas> G:=SL(2,R)
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas>  H:=PSL(2,R)
Value: disconnected split real group with Lie algebra 'sl(2,R)'
atlas>

Recall that \(H\cong SO(2,1)\).

Now, one of the main structural facts about a group is what its Cartan subgroups are. For that we can type the following sequence of commands:

atlas> nr_of_Cartan_classes (G)
Value: 2
atlas> set cartans =Cartan_classes (G)
Identifier cartans: [CartanClass]
atlas> #cartans
Value: 2
atlas> set T=cartans[0]
Identifier T: CartanClass
atlas>
atlas> T
Value: Cartan class #0, occurring for 2 real forms and for 1 dual real form
atlas>
atlas> print_Cartan_info (T)
compact: 1, complex: 0, split: 0
canonical twisted involution: e
twisted involution orbit size: 1; fiber size: 2; strong inv: 2
imaginary root system: A1
real root system: empty
complex factor: empty
atlas>

Wich gives us the number of conjugacy classes of Cartan subgroups of \(G\) and choosing one of those subgroups atlas gives more information about the Cartan. The function print_Cartan_info takes a Cartan class in \(G\) and provides the basic structural data of any representative in the Cartan class.

So this says that the first Cartan is a real cartan subgroup, meaning a connected complex torus defined over \(\mathbb R\). So, its real points form a real torus which can be written as a product of \((S^1)^a\), \(({\mathbb C}^{\times })^b\) and \(({\mathbb R}^{\times })^c\) factors. So, atlas gives the numbers (a,b,c). In this case the first Cartan has just a=1 circle factor, b=0 complex factors and c=0 real factors.

It also tells us the type of roots it has: imaginary, complex or real. Since the Cartan is compact, we only have imaginary roots. And these roots form a system of type A1.

Note that we also see that this Cartan occurs in different real forms of complex groups of type A1. The information about twisted involutions will be discussed later.

Now for information about the second Cartan subgroup:

atlas> set A=cartans[1]
Identifier A: CartanClass
atlas> A
Value: Cartan class #1, occurring for 1 real form and for 2 dual real forms
atlas> print_Cartan_info (A)
compact: 0, complex: 0, split: 1
canonical twisted involution: 1
twisted involution orbit size: 1; fiber size: 1; strong inv: 1
imaginary root system: empty
real root system: A1
complex factor: empty
atlas>

Here, the Cartan has just 1 copy of \({\mathbb R}^{\times }\) and no complex or circle factors. And it just has real roots of type A1. We will discuss later the information about involutions.

In contrast, the group \(SL(2,\mathbb C)\) has only one conjugacy class of cartans:

atlas> set G2=SL(2,C)
Identifier G2: RealForm (hiding previous one of type string)
atlas> nr_of_Cartan_classes (G2)
Value: 1
atlas> set cartans =Cartan_classes (G2)
Identifier cartans: [CartanClass] (hiding previous one of type [CartanClass])
atlas> cartans[0]
Value: Cartan class #0, occurring for 1 real form and for 1 dual real form
atlas>
atlas> print_Cartan_info (cartans[0])
compact: 0, complex: 1, split: 0
canonical twisted involution: e
twisted involution orbit size: 2; fiber size: 1; strong inv: 2
imaginary root system: empty
real root system: empty
complex factor: A1

Now for a larger group like \(Sp(4,\mathbb R)\), for example, we will have a compact cartan which is a product of two circles and all its roots are imaginary; a split cartan, that is, a product of \(({\mathbb R}^{\times })×({\mathbb R}^{\times })\) with all roots real; and two intermediate cartans; one complex isomorphic to \({\mathbb C}^{\times }\). This is sometimes called the short root Cartan. This is the one associated to a Levi factor \(Gl(2)\). Finally, the other Cartan is isomorphic to \(S^1×{\mathbb R}^{\times }\). The distinction between these two Cartans is subtle. Locally they are both isomorphic rank one Cartans and look like \(S^1×{\mathbb R}^{\times }\). But, one is \({\mathbb C}^{\times }\) and atlas can distinguish the two.

The root systems of these intermediate Cartans also transform accordingly. For the Compact Cartan we have an imaginary root system of type C2:

atlas> set G1=Sp(4,R)
Identifier G1: RealForm
atlas> G1
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> nr_of_Cartan_classes (G1)
Value: 4
atlas> set cartans =Cartan_classes (G1)
Identifier cartans: [CartanClass] (hiding previous one of type [CartanClass])
atlas>
atlas> print_Cartan_info (cartans[0])
compact: 2, complex: 0, split: 0
canonical twisted involution: e
twisted involution orbit size: 1; fiber size: 4; strong inv: 4
imaginary root system: C2
real root system: empty
complex factor: empty

Now for the most split Cartan, the last one, all of the roots are real:

atlas> print_Cartan_info (cartans[3])
compact: 0, complex: 0, split: 2
canonical twisted involution: 2,1,2,1
twisted involution orbit size: 1; fiber size: 1; strong inv: 1
imaginary root system: empty
real root system: C2
complex factor: empty
atlas>

For the complex intermidiate Cartan, we have an imaginary root system and a real root system, both of type A1:

atlas> cartans[1]
Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form
atlas> print_Cartan_info (cartans[1])
compact: 0, complex: 1, split: 0
canonical twisted involution: 2,1,2
twisted involution orbit size: 2; fiber size: 1; strong inv: 2
imaginary root system: A1
real root system: A1
complex factor: empty
atlas>

Lastly, the other intermidiate Cartan has also an imaginary and a real root system of type A1:

atlas> cartans[2]
Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms
atlas> print_Cartan_info (cartans[2])
compact: 1, complex: 0, split: 1
canonical twisted involution: 1,2,1
twisted involution orbit size: 2; fiber size: 2; strong inv: 4
imaginary root system: A1
real root system: A1
complex factor: empty
atlas>

So the distinction between these last two is burried in the extra information. More about this later.