Tutorial

Example \(Sp(6,\mathbb R)\)

Let us find the discrete series for this group:

atlas> G:=Sp(6,R)
Value: connected split real group with Lie algebra 'sp(6,R)'
atlas> set F=distinguished_fiber (G)
Variable F: [int]
atlas> F
Value: [0,1,2,3,4,5,6,7]
atlas>

atlas> ds:=all_discrete_series (G,rho(G)) Value: [final
parameter(x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=2,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=3,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=4,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=6,lambda=[3,2,1]/1,nu=[0,0,0]/1),final
parameter(x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1)]

atlas> show(ds)
final parameter(x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=2,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=3,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=4,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=5,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=6,lambda=[3,2,1]/1,nu=[0,0,0]/1)
final parameter(x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1)

Again, like in the case of \(Sp(4,\mathbb R)\) we can try to write each parameter in terms of a single x that has the usual compact simple roots. By looking at each x.

A more efficient way to do this is the following:

atlas> void: for i in F do prints(i," ",rho_K(KGB(G,i))) od
0 [ 1, 1, 0 ]/1
1 [ 1, 1, 0 ]/1
2 [ 1, 0, 1 ]/1
3 [ 1, 1, 0 ]/1
4 [ 1, 1, 0 ]/1
5 [  1,  0, -1 ]/1
6 [ 1, 0, 1 ]/1
7 [  1,  0, -1 ]/1
atlas>

We have two choices of x with the standard rho: x=5 and x=7. We choose one:

atlas> x_b:=KGB(G,5)
Value: KGB element #5
atlas> simple_roots(K_0(x_b))
Value:
|  1,  0 |
| -1,  1 |
|  0, -1 |

atlas>

atlas> void: for p in ds do prints(x(p), " ", hc_parameter (p, x_b)) od
KGB element #0 [  3,  1, -2 ]/1
KGB element #1 [  2,  1, -3 ]/1
KGB element #2 [  3,  2, -1 ]/1
KGB element #3 [  3, -1, -2 ]/1
KGB element #4 [  2, -1, -3 ]/1
KGB element #5 [ 3, 2, 1 ]/1
KGB element #6 [  1, -2, -3 ]/1
KGB element #7 [ -1, -2, -3 ]/1
atlas>

These lambdas are the conjugates of rho which are \(K\)-dominant. That is, modulo \(W_K\). They are all decreasing. So they are the usual Harish-Candra parameters for the eight discrete series of \(Sp(6,\mathbb R)\).

Now, as for previous examples we can write:

atlas> p:=discrete_series (x_b,[-1,-2,-3])
Value: final parameter(x=7,lambda=[3,2,1]/1,nu=[0,0,0]/1)
atlas>

So atlas knows what this is and makes lambda dominant and conjugates x_b to x_7.