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
\(\lambda=(1,-1)\):
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:
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:
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:trueatlas>equivalence_class_of(P)Value:[KGBelement#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 parabolics.at on the atlasLibrary 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 \(L\):
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 \(\gamma+\rho(\mathfrak u)\):
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 induction.at on the atlasLibrary 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)Runtimeerror:LisnotLeviofrealparabolic(incallatbasic.at:8:57-71oferror@string,built-in)[b=false,message="L is not Levi of real parabolic"]...(outputtruncated)
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).