General Parameters \(Sp(4,\mathbb R )\)¶

Let us look at all the representations of \(Sp(4,\mathbb R )\) with infinitesimal character rho

atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set B=block_of (trivial(G))
Variable B: [Param]
atlas> print_block(trivial(G))
Parameter defines element 10 of the following block:
 0:  0  [i1,i1]   1   2   ( 4, *)  ( 5, *)  *(x= 0,lam_rho=  [0,0], nu=  [0,0]/1)  e
 1:  0  [i1,i1]   0   3   ( 4, *)  ( 6, *)  *(x= 1,lam_rho=  [0,0], nu=  [0,0]/1)  e
 2:  0  [ic,i1]   2   0   ( *, *)  ( 5, *)  *(x= 2,lam_rho=  [0,0], nu=  [0,0]/1)  e
 3:  0  [ic,i1]   3   1   ( *, *)  ( 6, *)  *(x= 3,lam_rho=  [0,0], nu=  [0,0]/1)  e
 4:  1  [r1,C+]   4   9   ( 0, 1)  ( *, *)  *(x= 4,lam_rho=  [0,0], nu= [1,-1]/2)  1^e
 5:  1  [C+,r1]   7   5   ( *, *)  ( 0, 2)  *(x= 5,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
 6:  1  [C+,r1]   8   6   ( *, *)  ( 1, 3)  *(x= 6,lam_rho=  [0,0], nu=  [0,1]/1)  2^e
 7:  2  [C-,i1]   5   8   ( *, *)  (10, *)  *(x= 7,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
 8:  2  [C-,i1]   6   7   ( *, *)  (10, *)  *(x= 8,lam_rho=  [0,0], nu=  [2,0]/1)  1x2^e
 9:  2  [i2,C-]   9   4   (10,11)  ( *, *)  *(x= 9,lam_rho=  [0,0], nu=  [3,3]/2)  2x1^e
10:  3  [r2,r1]  11  10   ( 9, *)  ( 7, 8)  *(x=10,lam_rho=  [0,0], nu=  [2,1]/1)  1^2x1^e
11:  3  [r2,rn]  10  11   ( 9, *)  ( *, *)  *(x=10,lam_rho=  [1,1], nu=  [2,1]/1)  1^2x1^e
atlas>

There are 12 representations in the block of the trivial of \(Sp(4,\mathbb R)\). The first four and the last two are the ones we already know. Namely, the four discrete series and the two minimal principal series of \(G\). We know the first four are discrete series because the roots involved (third column of the above table) are all imaginary. Also, the second column tells us that their lengts are all zero. On the other hand the last two correspond to the KGB element x=10 and all the roots are real. So, this tells us we are in the split Cartan subgroup and these are minimal principal series.

Now we want to look at the representations numbered 4 to 9. For example:

atlas> p:=B[5]
Value: final parameter(x=5,lambda=[2,1]/1,nu=[0,1]/1)
atlas>
atlas> infinitesimal_character (p)
Value: [ 2, 1 ]/1
atlas>
atlas> set x=x(p)
Variable x: KGBElt
atlas> x
Value: KGB element #5
atlas> infinitesimal_character (p)
Value: [ 2, 1 ]/1
atlas> involution(x)
Value:
| 1,  0 |
| 0, -1 |

atlas>

This is the Cartan involution for one of the intermediate Cartan subgroups. Namely the “\(SL(2)\)- Cartan” or the “long root” Cartan. The Levi factor is \(SL(2,\mathbb R )\times GL(1,\mathbb R )\).

More about this later. Let us look at the Cartan involution for this Cartan subgroup

atlas> set theta=involution(x)
Variable theta: mat
atlas> theta
Value:
| 1,  0 |
| 0, -1 |

atlas>

Note that the \((-1)\)-eigenspace of this involution is the span of the second coordinate, which is where the``nu`` parameter lives:

atlas> nu(p)
Value: [ 0, 1 ]/1
atlas>

And lambda has to do with the restriction to the compact part of the Cartan subgroup of the infinitesimal character.

Cuspidal Data¶

To get more information about the above representation we use a new command

atlas> whattype cuspidal_data ?
Overloaded instances of 'cuspidal_data'
  Param->(([int],KGBElt),Param)
atlas>
atlas> set (P,sigma)=cuspidal_data (p)
Variable P: ([int],KGBElt)
Variable sigma: Param
atlas>
atlas> P
Value: ([1],KGB element #7)
atlas> sigma
Value: final parameter(x=0,lambda=[1,2]/1,nu=[1,0]/1)
atlas>

This is the cuspidal data of the representation with parameter p. That is, \(P\) is a real parabolic subgroup, sigma is a discrete series representation of the Levi factor \(M\) of \(P\) and the standard representation

\[I(p)=Ind_P ^G (\sigma )\]

Note that \(P\) is a pair of a string of integers and a KGB element. This is a generalization of a Borel and the string of integers lists the roots on the Levi factor. In this case it is just root number 1. The KGB element information has to do with \(K\) orbits on \(G/P\), which is a quotient of \(K\) orbits on \(G/B\). There is more information about this in the papers section of Atlas of Lie Groups

Now, let us find out a bit more about \(M\)

atlas> set M=Levi(P)
Variable M: RealForm
atlas> M
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'

This does not completely determines \(M\) since both,

\(SL(2,\mathbb R )\times GL(1, \mathbb R )\) and \(GL(2,\mathbb R )\) have the same Lie algebra. However, atlas has stored more information about \(M\)

atlas> simple_roots (M) Value: | 0 | | 2 |

atlas>

This information determines \(M\). Remember that these are the simple roots of \(SL(2,\mathbb R)\) whereas

atlas> H:=GL(2,R)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas>
atlas> simple_roots (H)
Value:
|  1 |
| -1 |

atlas>

Now about the representation sigma, remember that sigma is a parameter for \(M\), even though the parameters lambda and nu are in terms of the Cartan for \(Sp(4,\mathbb R )\)

atlas> sigma
Value: final parameter(x=0,lambda=[1,2]/1,nu=[1,0]/1)
atlas>
atlas> real_form (sigma)
Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)'
atlas>

That is, this is a parameter for a “relative” discrete series of \(M\). That is, a discrete series modulo the center. And \(M=SL(2,\mathbb R )\times GL(1,\mathbb R )\), where the simple factor corresponds to the root [0,2]. And \(GL(1,\mathbb R )\) corresponds to the first coordinate. Hence the nu=[1,0] is on the \(GL(1, \mathbb R )\)-factor.

Now the lambda is giving you the discrete series parameter of the Levi factor. And the infinitesimal character is

atlas> infinitesimal_character (sigma)
Value: [ 1, 2 ]/1
atlas>

This is normalized induction, so the infinitesimal character does not change. This is a conjugate of

atlas> rho(G)
Value: [ 2, 1 ]/1
atlas>

On the other hand

atlas> rho (M)
Value: [ 0, 1 ]/1
atlas>

atlas> hc_parameter (sigma)
Value: [ 1, 2 ]/1
atlas>