Tutorial

\(A_{\mathfrak q}(\lambda)\) Construction

An alternate way to define an \(A_{\mathfrak q}(\lambda)\) module is by specifying a KGB element (attached to the fundamental Cartan), a weight \(\lambda_q\) to define the \(\theta\)-stable Cartan, and the weight \(\lambda\) specifying the one-dimensional representation on \(L\). For this construction, the weight \(\lambda\) must satisfy that \(\lambda-\rho(\mathfrak u)\) is integral, and of course, it must be orthogonal to the roots of \(L\).

Let’s look at some examples in \(G=U(2,2)\). A convenient choice for x is KGB element 2, and we consider \(A_{\mathfrak q}(\lambda)\) modules attached to a \(\theta\)-stable parabolic with Levi factor \(U(2,1)\times U(0,1)\):

atlas> G:=U(2,2)
Value: connected quasisplit real group with Lie algebra 'su(2,2).u(1)'
atlas> x:=KGB(G,2)
Value: KGB element #2

atlas> set lamq=[1,1,1,0]
Variable lamq: [int]
atlas> P:=parabolic(lamq,x)
Parabolic is theta-stable.
Value: ([0,1],KGB element #2)
atlas> rho_u(P)
Value: [  1,  1,  1, -3 ]/2

Since \(\rho(\mathfrak u)\) is half-integral, we must choose \(\lambda\) to be half-integral as well:

atlas> set M1=Aq(x,[1,1,1,-1]/2,lamq)
Variable M1: Param
atlas> M1
Value: final parameter (x=15,lambda=[3,1,-1,-1]/2,nu=[1,0,-1,0]/1)

atlas> goodness(x,[1,1,1,-1]/2,lamq)
Weakly good

The function Aq(x,lam,lamq) computes \(\mathcal R_{\mathfrak q}(\mathbb C_{\lambda})\), but with a different normalization; there is a shift of \(\rho(\mathfrak u)\) so that the functor preserves infinitesimal characters: the resulting module shares the infinitesimal character with the one-dimensional representation \(\mathbb C_{\lambda}\) of (possibly a double cover of) \(L\). One advantage of this normalization is that it is easy to see whether \(\lambda\) is in the weakly fair range for \(\mathfrak u\): it must be weakly dominant:

atlas> goodness(x,[1,1,1,1]/2,lamq)
Weakly fair

atlas> goodness(x,[1,1,1,3]/2,lamq)
None

Let’s look at another example; this is discussed in Chapter 9 of Knapp-Vogan, “Cohomological Induction and Unitary Representations”. Here \(G=SO(5,4)\), and \(P\) is the unique \(\theta\)-stable parabolic with Levi factor \(U(2,2)\):

atlas> set G=SO(5,4)
Variable G: RealForm
atlas> set x=KGB(G,5)
Variable x: KGBElt
atlas> set lamq=[1,1,1,1]
Variable lamq: [int]

atlas> set P=parabolic(lamq,x)
Parabolic is theta-stable.
Variable P: ([int],KGBElt)
atlas> P
Value: ([0,1,2],KGB element #5)
atlas> rho_u(P)
Value: [ 2, 2, 2, 2 ]/1

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

We can construct the good \(A_{\mathfrak q}(\lambda)\) at infinitesimal character \(\rho\) using the two methods learned; let’s do that, just to check and confirm:

atlas> theta_induce_irreducible(trivial(L),G)
Value:
1*final parameter (x=43,lambda=[7,5,3,1]/2,nu=[3,1,-1,-3]/2)

atlas> Aq(x,[2,2,2,2],lamq)
Value: final parameter (x=43,lambda=[7,5,3,1]/2,nu=[3,1,-1,-3]/2)

Notice that our \(\lambda=(2,2,2,2)\) could also serve to define the parabolic; in this case, we could have omitted the additional entry lamq:

atlas> Aq(x,[2,2,2,2])
Value: final parameter (x=43,lambda=[7,5,3,1]/2,nu=[3,1,-1,-3]/2)

If we now move to the edge of the weakly fair range, Knapp/Vogan predict that the module will be reducible. The command Aq(x,lam,lamq) returns a parameter PROVIDED that the module is irreducible and nonzero:

atlas> Aq(x,[0,0,0,0],lamq)
Runtime error:
Aq is not irreducible. Use Aq_reducible(x,lambda) instead
(in call at basic.at:8:57-71 of error@string, built-in)
...(output truncated)

Since the module is reducible, we need to use the command Aq_reducible instead:

atlas> Aq_reducible(x,[0,0,0,0],lamq)
Value:
1*final parameter (x=84,lambda=[7,7,1,1]/2,nu=[3,3,0,0]/2)
1*final parameter (x=101,lambda=[7,7,3,3]/2,nu=[3,3,1,1]/2)

This weakly fair \(A_{\mathfrak q}(\lambda)\) module is indeed reducible, with two constituents.

Similarly, if our \(A_{\mathfrak q}(\lambda)\) module is zero, the command Aq(x,lam,lamq) will return an error message. Here is an example in \(Sp(4,\mathbb R)\):

atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> x:=KGB(G,2)
Value: KGB element #2
atlas> lam:=[0,0]
Value: [0,0]
atlas> lamq:=[2,1]
Value: [2,1]
atlas> goodness(x,lam,lamq)
Value: "Weakly good"
atlas> Aq(x,lam,lamq)
Runtime error:
  index 0 out of range (0<= . <0) in subscription P[0]
  [P=[]]
  ...(output truncated)

The parabolic has compact Levi factor, and the module is zero because there is a compact simple root that is orthogonal to \(\lambda\). In this case as well, the command Aq_reducible yields a nicer answer:

atlas> Aq_reducible(x,lam,lamq)
Value: Empty sum of standard modules