Real Parabolic Induction¶
In order to calculate some induced representations using atlas
, we must
first define parabolic subgroups. The corresponding atlas
data type is
Parabolic
, or, equivalently, KGPElt
, and consists of a pair (S,x)
,
where S
is a list of integers corresponding to the set of simple roots which
determine the conjugacy class of complex parabolics, and x
is a KGB
element. For some details, see the summary for the script file parabolics.at
on the atlas Library
page.
Let’s do an example to see how this works.
Defining a Real Parabolic Subgroup¶
Suppose we want to define a real parabolic subgroup of \(G=Sp(4,\mathbb R)\)
with Levi factor \(L\cong SL(2,\mathbb R)\times\mathbb R^{\times}\). This
Levi is attached to the long (simple) root in \(G\) which in atlas
is
numbered 1. In order to obtain a real parabolic subgroup, we choose x
to
to be the KGB
element associated to the maximally split Cartan, which
you may remember is the last element in KGB(G)
:
atlas> set G=Sp(4,R)
Variable G: RealForm
atlas> #KGB(G)
Value: 11
atlas> set x=KGB(G,10)
Variable x: KGBElt
atlas> set P=Parabolic:([1],x)
Variable P: ([int],KGBElt)
atlas> P
Value: ([1],KGB element #10)
To be sure that we have done this correctly, now check whether the parabolic we have defined is indeed real, and that the Levi factor is the one we wanted:
atlas> is_parabolic_real(P)
Value: true
atlas> set L=Levi(P)
Variable L: RealForm
atlas> L
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas> simple_roots(L)
Value:
| 0 |
| 2 |
An alternate way to define a (real) parabolic subgroup of a real group \(G\) is using a weight \(\lambda\). The simple roots orthogonal to the weight determine the type of the (underlying complex) parabolic. For the resulting parabolic subgroup to be real, the weight \(\lambda\) must satisfy \(\theta_x(\lambda)=-\lambda\):
atlas> set lambda=[1,0]
Variable lambda: [int]
atlas> set Q=parabolic(lambda,x)
Parabolic is real.
Variable Q: ([int],KGBElt)
atlas> Q
Value: ([1],KGB element #10)
Notice that if you define a parabolic subgroup using this command, atlas
will tell you what kind of parabolic you have: real, theta-stable, or neither.
Let’s check that we have defined the same parabolic as before:
atlas> P=Q
Value: true
Real Induction¶
Real parabolic induction in atlas
is normalized; the
infinitesimal character is preserved.
Now that we have a real parabolic subgroup of \(G\), let’s compute an induced representation. First we need to choose and write down a representation, i.e., a parameter, for the Levi subgroup \(L\). For example, let’s take the trivial representation:
atlas> set t=trivial(L)
Variable t: Param
atlas> t
Value: final parameter (x=2,lambda=[0,1]/1,nu=[0,1]/1)
Keep in mind how \(L\) is embedded in \(G\); the first coordinate
of lambda and nu corresponds to the \(\mathbb R^{\times}\) factor, the
second is the parameter for \(SL(2,\mathbb R)\). You can see KGB
for
\(L\):
atlas> print_KGB(L)
kgbsize: 3
Base grading: [1].
0: 0 [n] 1 2 (0,0)#0 e
1: 0 [n] 0 2 (0,1)#0 e
2: 1 [r] 2 * (0,0)#1 1^e
There are two ways in atlas
to induce a representation on \(L\) to
\(G\). The command real_induce_standard
takes the parameter for
\(L\) to represent a standard module, and writes the answer as a standard
module for \(G\):
atlas> real_induce_standard(t,G)
Value: final parameter (x=10,lambda=[2,1]/1,nu=[0,1]/1)
If you start with a single parameter, the output will be a single parameter
as well. You can also apply this function to a ParamPol
. Probably of more
interest will be the command real_induce_irreducible
which takes the
parameter of \(L\) to represent an irreducible representation, and returns
the composition series of the induced representation of \(G\):
atlas> real_induce_irreducible(t,G)
Value:
1*final parameter (x=4,lambda=[1,0]/1,nu=[1,-1]/2)
1*final parameter (x=10,lambda=[2,1]/1,nu=[1,0]/1)
You see that this induced representation is reducible, with two pieces.
Let’s look at a second example: this time, let’s take the real parabolic with Levi factor \(GL(2,\mathbb R)\) and compute the representation of \(G\) obtained by inducing the trivial on this Levi:
atlas> Q:=([0],x)
Value: ([0],KGB element #10)
atlas> L:=Levi(Q)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
Although the description of \(L\) is the same as in our first example, it is a different group:
atlas> simple_roots(L)
Value:
| 1 |
| -1 |
atlas> print_KGB(L)
kgbsize: 2
Base grading: [1].
0: 0 [n] 0 1 (0,0)#0 e
1: 1 [r] 1 * (0,0)#1 1^e
These are indeed the data for \(GL(2,\mathbb R)\). Now let’s induce:
atlas> t:=trivial(L)
Value: final parameter (x=1,lambda=[1,-1]/2,nu=[1,-1]/2)
atlas> real_induce_irreducible(t,G)
Value:
1*final parameter (x=10,lambda=[2,1]/1,nu=[1,1]/2)
So this time, the induced representation is irreducible.