Principal Series of Split Groups¶

So far the focus has been on Cartan subgroups, whose information is encoded on the element x as the Cartan involution of the complex abstract group \(H\), which determines the real group \(H(\mathbb R)\). Now x is really a \(K\)-orbit on \(G/B\). So, it is the support of the corresponding \(D\)-module. In order to explain this in detail we will look at some easy cases. In particular we will be talking about the principal series of split groups.

So, we start with a group \(G\) and a parameter p=(x, lambda, nu) where x encodes the above information and \(\lambda \in X^* /(1-\theta )X^*\) and \(\nu \in {X^* \otimes \mathbb Q }^{-\theta}\). With these data we obtain a character of \(H(\mathbb R)\) with differential \({(1+\theta )\over 2}\lambda + \nu\).

From this character we get a representation of the group G.

Minimal Principal Series for split groups¶

If \(H\) is the split Cartan of \(G\). Let \(B\) be a borel including this Cartan. We can construct the induced representation \(Ind_B ^G (\chi)\) where \(\chi\) is a character of \(H(\mathbb R)\).

For example, if \(G=SL(2, \mathbb R )\) we can look again at the list of parameters with infinitesimal character \(\rho\).

Recall the command all_parameters_gamma takes a real form and an ifinitesimal character, which is a rational vector, and gives you a list of parameters for the representations of the real form with that infinitesimal character:

atlas> whattype all_parameters_gamma ?
Overloaded instances of 'all_parameters_gamma'
  (RealForm,ratvec)->[Param]
atlas> set G=SL(2,R)
Variable G: RealForm
atlas> G
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas> rho(G)
Value: [ 1 ]/1
atlas> set parameters=all_parameters_gamma (G,[1])
Variable parameters: [Param]
atlas> #parameters
Value: 4
atlas> void: for p in parameters 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>

Here the x is giving us Cartan involutions of the Cartans:

atlas> involution(KGB(G,0))
Value:
| 1 |

atlas> involution(KGB(G,1))
Value:
| 1 |

atlas> involution(KGB(G,2))
Value:
| -1 |

So, the first two parameters in the list are associated to the compact Cartan; the last two to the split one.

We can find out more about the Cartan for each parameter p as follows:

atlas> set p= parameters[3]
Identifier p: Param (hiding previous one of type Param)
atlas> p
Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>
atlas> x:=x(p)
Value: KGB element #2
atlas> set H=Cartan_class(x)
Identifier H: CartanClass (hiding previous one of type RealForm)
atlas> print_Cartan_info(H)
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>

So, this is the split Cartan for this group, with one real factor and no compact or complex factor. We can ignore the rest of the information for the moment.

As we said above, the last two in the list of parameters for \(G\) are the ones associated to this split Cartan subgroup; namely the two principal series with parameter nu=1:

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

The difference between these two are the lambda. The trivial representation of \(G\) is:

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

So, the parameter is the induced representation that has the trivial as its irreducible quotient. This is the spherical principal series. There is a rho shift for the lambda so that the spherical pricipal series has lambda=1 instead of 0 as you might expect. The other principal series is the non spherical irreducible:

atlas> set ps2=parameters[3]
Identifier ps2: Param
atlas> ps2
Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1)
atlas>
atlas> set std=I(ps2)
Identifier I: (Param,string)
atlas> std
Value: (final parameter (x=2,lambda=[2]/1,nu=[1]/1),"std")
atlas>
atlas> show(composition_series (std))
1*J(x=2,lambda=[2/1],nu=[1/1])
atlas>

Here J stands for an irreducible representation and the single line above says that there is only one composition factor in this representation. Namely, the irreducible principal series itself.

On the other hand, the composition factors of the spherical principal series are:

atlas> set ps1=parameters[2]
Identifier ps1: Param (hiding previous one of type Param)
atlas>
atlas> show(composition_series (I(ps1)))
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>

This standard module defined by the above parameter has three composition factors, all irreducible. So I(ps1) is the sum in the Grothendieck group of three irreducible composition factors.

Similarly, if we take parameters of a spherical representation with non-integral infinitesimal character we get irreducibility:

atlas> x
Value: KGB element #2
atlas> set q=parameter (x, [1], [3/2])
Identifier q: Param (hiding previous one of type Param)
atlas> infinitesimal_character (q)
Value: [ 3 ]/2
atlas> show(composition_series (I(q)))
1*J(x=2,lambda=[1/1],nu=[3/2])
atlas>
atlas> set q=parameter (x, [0], [3/2])
Identifier q: Param (hiding previous one of type Param)
atlas> show(composition_series (I(q)))
1*J(x=2,lambda=[2/1],nu=[3/2])
atlas>

So there are two large families of irreducible principal series, one with parameters of the form (x, [1], nu), and the other with parameters x, [0], nu ), where ``nu is non-integral:

Cuspidal Data¶

Another thing you can do is get also information about cuspidal data used to construct this representation:

set p=parameter(x,[2],[3/2])
Identifier p: Param (hiding previous one of type Param)
atlas> set (P,q)=cuspidal_data(q)
Identifiers P: ([int],KGBElt), q: Param (hiding previous one of type Param)
atlas> Levi(P)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> q
Value: final parameter (x=0,lambda=[1]/1,nu=[1]/1)
atlas> p
Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1)

So \(P\) is a parabolic whith Levi factor \(GL(1,\mathbb R)\) and q is acharacter of \(GL(1,\mathbb R)\). So we can extract the character of the Cartan by finding the Cuspidal data for the representation with parameter p:

atlas>
atlas> real_form(q)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas> Levi(P)
Value: disconnected split real group with Lie algebra 'gl(1,R)'
atlas>

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

Aside¶

NOTE: WHAT FOLLOWS IN THE NEXT LINES IS NOT COMPLETED. I NEED TO WORK A BIT MORE ON WHAT ALL THESE COMMANDS DO. THIS WAS NOT DISCUSSED IN THE VIDEOS AT THIS POINT. BUT I THINK IT WOULD BE GOOD TO HAVE A DISCUSSION HERE ABOUT THESE COMMANDS

Other possible commands are:

theta_induce_irreducible
theta_induce_irreducible_as_sum_of_standards
theta_induce_standard
theta_stable_data
theta_stable_parabolic
theta_stable_parabolics
theta_stable_parabolics_type

For example:

atlas> theta_stable_data (p)
Value: (([0],KGB element #2),final parameter (x=2,lambda=[2]/1,nu=[3]/2))
atlas>

In this case the Levi is all of \(G\). So this says that the representation is induced from \(G\) to \(G\).

To go back use theta_induced_standard

Example \(G=PSL(2,\mathbb R)\)¶

Another group we can look at is:

atlas> G:PSL(2,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> G
Value: disconnected split real group with Lie algebra 'sl(2,R)'
atlas>

Here the complex Lie group is \(G(\mathbb C )=PSL(2,\mathbb C )=SL(2,\mathbb C)/{\pm 1}\). Its real points are disconnected, and they are the group \(PSL(2, \mathbb R ) \cong SO(2,1)\):

atlas> rho(G)
Value: [ 1 ]/2
atlas> set parameters=all_parameters_gamma (G,rho(G))
Variable parameters: [Param] (overriding previous instance, which had type [Param])
atlas>

Note we can use rho(G) instead of the vector value for \(\rho\). The parameters for this group are almost like those for \(SL(2,\mathbb R)\), except that the Weyl group of the compact Cartan has changed and the number of parameters is now just three:

atlas> #parameters
Value: 3
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[1]/2,nu=[0]/1)
final parameter (x=1,lambda=[1]/2,nu=[1]/2)
final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas>

We still have two principal series with infinitesimal character \(\rho\). But we now only have one discrete series representation associated to the compact Cartan, namely the sum of the two discrete series for \(SL(2,\mathbb R)\) are now a single irreducible representation of \(PSL(2, \mathbb R )\).

Now let us look at the trivial representation

atlas> p:trivial(G)
Variable p: Param
atlas> p
atlas> dimension (p)
Value: 1
atlas>

One thing to have in mind is that the trivial representation is always given by the maximal number x and lambda=nu=rho

This parameter has composition series:

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

Actually it is best to use the command show(composition_series(I(p))))

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

So, this induced representation for \(PSL(2,\mathbb R )\) has two factors: the trivial representation (with x=1 and \(\lambda=\nu=\rho\) ) and a discrete series (with x=0).

What is the other principal series attached to the split Cartan? For \(SL(2,\mathbb R )\) the other representation attached to the split Cartan was an infinite demensional irreducible principal series. However here we have:

atlas> q:parameters[2]
Variable q: Param
atlas> q
Value: final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas> is_finite_dimensional (q)
Value: true
atlas> p=q
Value: false
atlas>
atlas> p
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/2)
atlas> q
Value: final parameter (x=1,lambda=[3]/2,nu=[1]/2)
atlas>

This is another one dimensional representation of G not equivalent to the trivial representation. Recall that \(PSL (2,\mathbb R )\) is disconnected, so q is the parameter for the sign representation. In other words the standard module attached to this parameter is a principal series which has as its unique irreducible quotient the sign representation of \(PSL (2,\mathbb R )\).

Now for another example:

atlas> set p=parameter(KGB(G,1),[1]/2,[1])
Variable p: Param
atlas> p
Value: final parameter (x=1,lambda=[1]/2,nu=[1]/1)
atlas> show(composition_series (I(p)))
1*J(x=1,lambda=[1/2],nu=[1/1])
atlas>

So, the composition series gives an irreducible. Even though nu is an integer this is not an irreducibility point.

Example G=Sp(2,R)¶

Now lets find all representations with infinitesimal character \(\rho\)

atlas> G:Sp(4,R)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> G
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set parameters=all_parameters_gamma (G, rho(G))
Variable parameters: [Param]
atlas> rho(G)
Value: [ 2, 1 ]/1
atlas> #parameters
Value: 18
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[2,1]/1,nu=[0,0]/1)
final parameter (x=1,lambda=[2,1]/1,nu=[0,0]/1)
final parameter (x=2,lambda=[2,1]/1,nu=[0,0]/1)
final parameter (x=3,lambda=[2,1]/1,nu=[0,0]/1)
final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2)
final parameter (x=5,lambda=[2,1]/1,nu=[0,1]/1)
final parameter (x=5,lambda=[2,2]/1,nu=[0,1]/1)
final parameter (x=6,lambda=[2,1]/1,nu=[0,1]/1)
final parameter (x=6,lambda=[2,2]/1,nu=[0,1]/1)
final parameter (x=7,lambda=[2,1]/1,nu=[2,0]/1)
final parameter (x=7,lambda=[3,1]/1,nu=[2,0]/1)
final parameter (x=8,lambda=[2,1]/1,nu=[2,0]/1)
final parameter (x=8,lambda=[3,1]/1,nu=[2,0]/1)
final parameter (x=9,lambda=[2,1]/1,nu=[3,3]/2)
final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1)
atlas>

There are 18 representations with infinitesimal character \(\rho\). The last four parameters have \(K\backslash G/B\) element x=10. They correspond to the four priincipal series attached to the split Cartan.

More generally if \(G(\mathbb R)\) is split of rank \(n\), the number of minimal principal series of infinitesimal character rho corresponds to the set of characters of the split cartan \(({\mathbb R}^{\times}) ^n\) . That is the set {(lambda,nu)} and up to the Weyl group, nu=rho. So there are \(2^n\) choices of lambda which is a character of \(({\mathbb Z}_2)^n\).

In this case the rank is \(2\), so there are four, namely the last four in the above list.

Let us make a separate list for them:

atlas> set ps=[parameters[14],parameters[15],parameters[16],parameters[17]]
Variable ps: [Param]
atlas> void: for p in ps do prints(p) od
final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1)
final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1)
atlas>

These parameters are all principal series. How do we tell them apart?

Each one is giving a character of the split Cartan. They have the same nu and same x=10 and a different lambda. Each lambda is a character of \({\mathbb Z}_2 \times {\mathbb Z}_2\). In other words they are in \(X^*/2X^*\).

To know which is which we look at their tau invariant:

atlas> void: for p in ps do prints(p," ",tau(p)) od
final parameter (x=10,lambda=[2,1]/1,nu=[2,1]/1) [0,1]
final parameter (x=10,lambda=[3,1]/1,nu=[2,1]/1) [1]
final parameter (x=10,lambda=[2,2]/1,nu=[2,1]/1) []
final parameter (x=10,lambda=[3,2]/1,nu=[2,1]/1) [0]
atlas>

So the tau invariant is big for the element ps[0], which means the irreducible is a small representation. In fact that is the trivial representation. In contrast, the smallest tau invariant, the empty set, correspnding to the element ps[3] gives the biggest representation. In this case this is the irreducible principal series. The other two, namely the elements ps[2] and ps[4] in the list, correspond to the long and short roots respectively. So each of them are distinquished by their tau invariant.

Now lets look at the composition series of the standard module for the trivial rep:

atlas> p:ps[0]
Variable p: Param
atlas> show(composition_series(I(p)))
1*J(x=0,lambda=[2/1,1/1],nu=[0/1,0/1])
1*J(x=1,lambda=[2/1,1/1],nu=[0/1,0/1])
2*J(x=4,lambda=[2/1,1/1],nu=[1/2,-1/2])
1*J(x=5,lambda=[2/1,1/1],nu=[0/1,1/1])
1*J(x=6,lambda=[2/1,1/1],nu=[0/1,1/1])
1*J(x=7,lambda=[2/1,1/1],nu=[2/1,0/1])
1*J(x=8,lambda=[2/1,1/1],nu=[2/1,0/1])
1*J(x=9,lambda=[2/1,1/1],nu=[3/2,3/2])
1*J(x=10,lambda=[2/1,1/1],nu=[2/1,1/1])
atlas>

This standard module is the sum of all the above irreducibles with certain multiplicities. The last irreducible is the trivial representation. This is the biggest composition series. It is the most reducible principal series, which you can detect by its tau invariant.

On the other hand the empty tau invariant says that the representation is irreducible:

atlas> p:ps[2]
Variable p: Param (overriding previous instance, which had type Param)
atlas> show(composition_series(I(p)))
1*J(x=10,lambda=[2/1,2/1],nu=[2/1,1/1])
atlas>