Parameters

Introduction

The basic object in atlas is a parameter and the parameter space parametrizes both, the irreducible representations of a reductive algebraic group as well as the standard modules. In other words, for each p in the parameter space there are an irreducible module \(J(p)\) and a standard module \(I(p)\) associated to p. Namely \(I(p)\) is a representation induced from a limit of discrete series and things are set up, following Langlands classification, so that this standard module has a unique irrudicible quotient \(J(p)\). This quotient is also parametrized by the same parameter p.

So this parameter p is the basic object behind this classification theory.

In Adams’ and DuCloux’s paper, “Algorithms for representations of real groups”, Section 1, the authors use Langlands classification to describe the algorithm that will associate, to each real group, the parameter space in question.

More precisely, assume for the moment that rho exponentiates to a character of a torus in \(G(\mathbb C )\). Then, the representations of \(G(\mathbb R)\), with a fixed (regular) infinitesimal character lambda are parametrized by \(G(\mathbb R)\)-orbits of pairs \((H(\mathbb R ), \chi )\); where \(H(\mathbb R )\) is a Cartan subgroup of \(G(\mathbb R )\) and \(\chi\) is a character of \(H(\mathbb R )\) so that the differential of \(\chi\) equals lambda up to \(G(\mathbb C )\)-conjugacy.

Parameters for \(SL(2,\mathbb R)\)

Let’s look at \(G=SL(2,\mathbb R)\) and representations with infinitesimal character rho How many are there? We need to look at conjugacy classes of cartans and their characters.

Let’s review a few things we know about \(SL(2,\mathbb R)\):

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

atlas> root_datum (G)
Value: simply connected root datum of Lie type 'A1'
atlas> simple_roots(G)
Value:
| 2 |

atlas> rho(G)
Value: [ 1 ]/1

atlas> nr_of_Cartan_classes (G)
Value: 2

atlas> void: for H in Cartan_classes (G) do prints(H) od
Cartan class #0, occurring for 2 real forms and for 1 dual real form
Cartan class #1, occurring for 1 real form and for 2 dual real forms
atlas>

atlas> set T= Cartan_classes (G)[0]
Identifier T: CartanClass
atlas> T
Value: Cartan class #0, occurring for 2 real forms and for 1 dual real form
atlas> set A= Cartan_classes (G)[1]
Identifier A: CartanClass (hiding previous one of type mat)
atlas> A
Value: Cartan class #1, occurring for 1 real form and for 2 dual real forms
atlas>


atlas> occurrence_matrix (G)
Value:
| 1, 0 |
| 1, 1 |

atlas> void: for H in real_forms (G) do prints(H) od
compact connected real group with Lie algebra 'su(2)'
connected split real group with Lie algebra 'sl(2,R)'
atlas>

So, the split form of type A1 has two Cartans, the compact one, \(T=S^1\) and the split one, \(A={\mathbb R}^{\times }\).

Now, the characters for \(T\) are of the form \(e^{ik\theta}\) with \(k \in \mathbb Z\). The ones corresponding to rho are \(\{e^{i\theta }, e^{-i\theta }\}\) and they are not conjugate under the Weyl group of \(T\), since -1 is not in this Weyl group.

On the other hand, for \(A={\mathbb R}^{\times }\), the characters whose differential is equal to rho are \(\{ x\rightarrow x, x^{-1},|x|, |x|^{-1} : x\in A \}\), where \(|x|=sign(x)x\).

In this case -1 is in the Weyl group of \(A\). So, up to conjugacy, we have that \({\widehat A} \rightarrow \{ x, |x| \}\).

This says that we have exactly four representations of \(SL(2,\mathbb R)\) with infinitesimal character rho; two from each Cartan.

Let us look for those representations of \(SL(2,\mathbb R)\). The command parameters_gamma (G,[1]) looks for all the parameters of \(G\) with that infinitesimal character [1]:

atlas> set P=all_parameters_gamma (G,[1])
Identifier P: [Param]
atlas> #P
Value: 4
atlas>
atlas> void: for p in P do prints(p) od
final parameter (x=0,lambda=[1]/1,nu=[0]/1)
final parameter (x=1,lambda=[1]/1,nu=[0]/1)
final parameter (x=2,lambda=[1]/1,nu=[1]/1)
final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>

This is the set of parameters for representations of \(SL(2,\mathbb R)\) with infinitesimal character rho. Each parameter is a triple. (x, lambda, nu). We will explain each of these later. But for now we can say that the representation theory of \(SL(2,\mathbb R)\) tells us that there are four representations with infinitesimal character rho. Two of them are the discrete series associated to the compact Cartan and correspond to the two parameters above with nu=0; the other two are the trivial representation and an irreducible principal series; both, attached to the split Cartan and correspond to the parameters with nu=1.

We will say more about the representations of \(SL(2,\mathbb R)\) later. But, as it is illustrated here, the theory tells us we first need to understand the characters of Tori. We do this in the next section.