Cohomological Parabolic Induction

Defining \(\theta\)-stable parabolic subalgebras for a given real group \(G\) is a little trickier than defining real parabolic subgroups because there can be more that one \(K\) conjugacy class of such structures attached to a given (type of) complex parabolic subgroup.

Defining a \(\theta\)-Stable Parabolic Subalgebra

In the section on Real Parabolic Induction we discussed two different ways of defining a real parabolic subgroup: by giving the complex parabolic subgroup type (i.e., a subset of the simple roots), and by giving a weight \(\lambda\). Let’s focus here on the second technique, which may seem more natural and familiar. Since a \(\theta\)-stable parabolic subalgebra may be thought of as \(\mathfrak q(\lambda)=\mathfrak l(\lambda)+\mathfrak u(\lambda)\) for a weight \(\lambda\) in the compact part of the fundamental Cartan subalgebra, the first step is to choose a KGB element x attached to the fundamental Cartan. It also needs to be in the distinguished fiber (which is automatic in the equal rank case). In order to obtain a \(\theta\)-stable parabolic subalgebra, you need to choose a weight \(\lambda\) that is fixed by the involution \(\theta_x\). In the equal rank case, this is, of course, also automatic for this choice of x.

Let’s look at an example: Let \(G=Sp(4,\mathbb R)\) once again, and let’s choose x to be KGB element 2. This is the element attached to the holomorphic (or antiholomorphic) discrete series of \(G\), so that the simple root #0, \(e_1-e_2\), is compact. Then the unique \(\theta\)-stable parabolic subalgebra with Levi factor \(U(1,1)\) is the one attached to, for example, the weight \(\lambda=(1,-1)\):

atlas> set G=Sp(4,R)
Variable G: RealForm
atlas> set x=KGB(G,2)
Variable x: KGBElt

atlas> set P=parabolic([1,-1],x)
Parabolic is theta-stable.
Variable P: ([int],KGBElt)

atlas> P
Value: ([0],KGB element #0)

atlas> set L=Levi(P)
Variable L: RealForm
atlas> L
Value: connected quasisplit real group with Lie algebra 'sl(2,R).u(1)'

Notice that atlas prints a message that the parabolic is indeed \(\theta\)-stable. Notice also that atlas conjugates the weight \(\lambda\) and x simultaneously to make the weight dominant before calculating P; we could have defined this parabolic using the dominant weight \((1,1)\) and the KGB element #0:

atlas> set Q=parabolic([1,1],KGB(G,0))
Parabolic is theta-stable.
Variable Q: ([int],KGBElt)
atlas> P=Q
Value: true

If we use the weight \((1,1)\) with our original x to construct a parabolic, we get one with compact Levi factor:

atlas> set P2=parabolic([1,1],x)
Parabolic is theta-stable.
Variable P2: ([int],KGBElt)
atlas> P2
Value: ([0],KGB element #2)
atlas> Levi(P2)
Value: compact connected real group with Lie algebra 'su(2).u(1)'

and \(\lambda=(-1,-1)\) would give the opposite parabolic.

You can get a list of all \(\theta\)-stable parabolics for \(G\) using the command theta_stable_parabolics:

atlas> set tsp=theta_stable_parabolics(G)
Variable tsp: [([int],KGBElt)]

atlas> void: for P@i in tsp do prints(i," ",P) od
0 ([],KGB element #0)
1 ([],KGB element #1)
2 ([],KGB element #2)
3 ([],KGB element #3)
4 ([0],KGB element #2)
5 ([0],KGB element #3)
6 ([0],KGB element #4)
7 ([1],KGB element #5)
8 ([1],KGB element #6)
9 ([0,1],KGB element #10)

Notice that there are four \(\theta\)-stable parabolics corresponding to the empty set of simple roots, i.e., to the Borel, one for each discrete series. You can also get a list of the parabolics of a certain type only:

atlas> set tsp_0=theta_stable_parabolics_type(G,[0])
Variable tsp_0: [([int],KGBElt)]

atlas> void: for P@i in tsp_0 do prints(i," ",P) od
0 ([0],KGB element #2)
1 ([0],KGB element #3)
2 ([0],KGB element #4)

In this list, the KGB element given is the maximal element in the equivalence class on KGB(G) defined by the set of simple roots [0]; our parabolic P defined earlier is #2 in this list:

atlas> P=tsp_0[2]
Value: true
atlas> equivalence_class_of(P)
Value: [KGB element #0,KGB element #1,KGB element #4]

The equivalence class of P is the set of KGB elements obtained from x by cross actions and Cayley transforms through simple root #0 (in general, through any of the simple roots listed). Each of these KGB elements will define the same parabolic. (See the summary for the script file on the atlas Library page for more details.)

Theta-Stable Induction

The commands for theta-stable, or cohomological parabolic, induction work in a similar fashion to the corresponding commands in the real parabolic case. We can theta-induce standard modules (theta_induce_standard) or irreducibles, and the answers need to be understood in those terms. Let’s focus here on the second type of induction: inducing an irreducible representation of \(L\) to get the composition series of the resulting representation of \(G\).

Let’s stay with \(G=Sp(4,\mathbb R)\), and \(P\) the \(\theta\)-stable parabolic with Levi factor \(L=U(1,1)\). First take the trivial representation of \(L\):

atlas> t:=trivial(L)
Value: final parameter (x=2,lambda=[1,-1]/2,nu=[1,-1]/2)
atlas> set p=theta_induce_irreducible(t,G)
Variable p: ParamPol
atlas> p
1*final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2)Variable p: ParamPol

atlas> goodness(t,G)

This is of course an \(A_{\mathfrak q}(\lambda)\) module in the good range, and therefore, as expected, irreducible. Theta-induction takes representations of infinitesimal character \(\gamma\) to representations of infinitesimal character \(\gamma+\rho(\mathfrak u)\):

atlas> infinitesimal_character (t)
Value: [  1, -1 ]/2
atlas> infinitesimal_character (p)
Value: [ 2, 1 ]/1
atlas> rho_u(P)
Value: [ 3, 3 ]/2

The output is of type ParamPol. Next, let’s induce the one-dimensional \(det^{-1}\) of \(L\):

atlas> set p1=parameter(L,2,[-1,-3]/2,[-1,-3]/2)
Variable p1: Param
atlas> goodness(p1,G)
Weakly good
atlas> theta_induce_irreducible(p1,G)
1*final parameter (x=4,lambda=[1,0]/1,nu=[1,-1]/2)

Of course, we can choose any irreducible representation on \(L\) at all. For a non-unitary example, here is a finite dimensional representation:

atlas> set p=parameter(L,2,[1,-5]/2,[1,-5]/2)
Variable p: Param (overriding previous instance, which had type Param)
atlas> dimension(p)
Value: 3
atlas> goodness (p,G)

atlas> theta_induce_irreducible(p,G)
1*final parameter (x=4,lambda=[2,1]/1,nu=[1,-1]/2)
1*final parameter (x=9,lambda=[2,1]/1,nu=[3,3]/2)

This parameter is outside the fair range, and the induced representation is reducible. The calculation involves wall crossings and coherent continuation action. (See the summary for the script file on the atlas  Library page for more details.)

Notice that the induction functions will accept only parameters on Levi factors of the right kind of parabolics; entering a parameter on a Levi subgroup that does not come from a real parabolic subgroup will result in an error message:

atlas> real_induce_irreducible(t,G)
Runtime error:
L is not Levi of real parabolic
(in call at of error@string, built-in)
[b=false, message="L is not Levi of real parabolic"]
...(output truncated)

Similarly, the function theta_induce_irreducible requires the input of a parameter on a Levi subgroup coming from a \(\theta\)-stable parabolic subalgebra. Indeed, a Levi subgroup of \(G\) uniquely defines the parabolic it came from. The command make_parabolic(L,G) reverses the function Levi(P).