:math:`A_{\mathfrak q}(\lambda)` Construction
================================================

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

Let's look at some examples in :math:`G=U(2,2)`. A convenient choice for ``x``
is ``KGB`` element 2, and we consider :math:`A_{\mathfrak q}(\lambda)` modules
attached to a :math:`\theta`-stable parabolic with Levi factor
:math:`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 :math:`\rho(\mathfrak u)` is half-integral, we must choose
:math:`\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
:math:`\mathcal R_{\mathfrak q}(\mathbb C_{\lambda})`, but with a different
normalization; there is a shift of :math:`\rho(\mathfrak u)` so that the
functor preserves infinitesimal characters: the resulting module shares the
infinitesimal character with the one-dimensional representation
:math:`\mathbb C_{\lambda}` of (possibly a double cover of) :math:`L`. One
advantage of this normalization is that it is easy to see whether
:math:`\lambda` is in the weakly fair range for :math:`\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 :math:`G=SO(5,4)`, and :math:`P` is the unique :math:`\theta`-stable parabolic with Levi factor :math:`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 :math:`A_{\mathfrak q}(\lambda)` at infinitesimal
character :math:`\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 :math:`\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 :math:`A_{\mathfrak q}(\lambda)` module is indeed reducible,
with two constituents.

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