\(K\backslash G/B\) for other Cartan subgroupsΒΆ

Let us look at \(G=Sp(4,R)\). The \(K\backslash G/B\) elements are:

atlas> G
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> print_KGB (G)
kgbsize: 11
Base grading: [11].
 0:  0  [n,n]    1   2     4   5  (0,0)#0 e
 1:  0  [n,n]    0   3     4   6  (1,1)#0 e
 2:  0  [c,n]    2   0     *   5  (0,1)#0 e
 3:  0  [c,n]    3   1     *   6  (1,0)#0 e
 4:  1  [r,C]    4   9     *   *  (0,0) 1 1^e
 5:  1  [C,r]    7   5     *   *  (0,0) 2 2^e
 6:  1  [C,r]    8   6     *   *  (1,0) 2 2^e
 7:  2  [C,n]    5   8     *  10  (0,0)#2 1x2^e
 8:  2  [C,n]    6   7     *  10  (0,1)#2 1x2^e
 9:  2  [n,C]    9   4    10   *  (0,0)#1 2x1^e
10:  3  [r,r]   10  10     *   *  (0,0)#3 1^2x1^e
atlas>

Recall that the first four form the fundamental fiber that go to the Cartan subgroup 0, the compact one. Elements 5 through 8 are attached to Cartan subgroup number 2, etc. The last collumn tells us that the fiber attached to the involution 2^e consists of elements 5 and 6 and the fiber corresponding to the element 1x2^e are elements 7 and 8. Here 2^e is just Cayley transform by \({\alpha}_2\), whereas 1x2^e corresponds to conjugation by \({\alpha}_1\) composed with the Cayley transform by \({\alpha}_2\).

Let us recall which Cartan subgroups and Weyl groups correspond to each fiber:

atlas> set H=Cartan_class(G,0)
Variable H: CartanClass (overriding previous instance, which had type string (constant))
atlas> print_Cartan_info (H)
compact: 2, complex: 0, split: 0
canonical twisted involution: e
twisted involution orbit size: 1; fiber size: 4; strong inv: 4
imaginary root system: C2
real root system: empty
complex factor: empty
atlas>

As we know this is the Compact Cartan subgroup associated to the distinguished fiber:

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
   W^C is trivial
   A is trivial
   W_ic is a Weyl group of type A1
   W^R is trivial

   generators for W_ic:
   2,1,2
   atlas>

This is a Weyl group of type A1. So, the number of KGB orbits for this Cartan is 8/4=2

Now for one of the intermediate Cartan subgroups we have:

atlas> H:=Cartan_class(G,1)
Value: Cartan class #1, occurring for 2 real forms and for 1 dual real form
atlas>
atlas> print_Cartan_info (H)
compact: 0, complex: 1, split: 0
canonical twisted involution: 2,1,2
twisted involution orbit size: 2; fiber size: 1; strong inv: 2
imaginary root system: A1
real root system: A1
complex factor: empty
atlas>

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is an elementary abelian 2-group of rank 1
W_ic is trivial
W^R is a Weyl group of type A1

generators for A
1
generators for W^R:
2,1,2
atlas>

This is a copy of \({\mathbb C}^\times\) with Weyl group of order 4. So the number of KGB orbits is 8/4=2

Let us see what the \(W\)-orbit of one element is, say:

atlas> set x=KGB(G,4)
Variable x: KGBElt
atlas> void: for w in W do prints(cross(w,x)) od
KGB element #4
KGB element #4
KGB element #9
KGB element #9
KGB element #9
KGB element #9
KGB element #4
KGB element #4
atlas>

Starting with element 4 the order of its stabilizer has four elements. And if we list all the elements of \(W\):

atlas> void: for (,w) in W do prints(w) od
[]
[0]
[1]
[1,0]
[0,1]
[0,1,0]
[1,0,1]
[1,0,1,0]
atlas>

We see that the elements [], [0], [1,0,1], and [1,0,1,0] all stabilize element 4. So the order of the stabilizer is 4. Similarly, for element 9.

Now for the next Cartan subgroup:

atlas> H:=Cartan_class(G,2)
Value: Cartan class #2, occurring for 1 real form and for 2 dual real forms
   atlas>
   atlas> print_Cartan_info (H)
   compact: 1, complex: 0, split: 1
   canonical twisted involution: 1,2,1
   twisted involution orbit size: 2; fiber size: 2; strong inv: 4
   imaginary root system: A1
   real root system: A1
   complex factor: empty
   atlas>

This subgroup has order four. And its real Weyl group has order 2:

atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is trivial
W^R is a Weyl group of type A1

generators for W^R:
1,2,1
atlas>

Then the number of KGB orbits is 8/2=4 and we can verify also that each stabilizer is order 2:

atlas> x:=KGB(G,5)
Variable x: KGBElt
atlas>
atlas> void: for w in W do prints(cross(w,x)) od
KGB element #5
KGB element #7
KGB element #5
KGB element #8
KGB element #7
KGB element #6
KGB element #8
KGB element #6
atlas>
atlas> void: for (,w) in W do prints(w) od
[]
[0]
[1]
[1,0]
[0,1]
[0,1,0]
[1,0,1]
[1,0,1,0]
atlas>

Now for completeness, let us look at the split Cartan subgroup:

atlas> H:=Cartan_class(G,3)
Value: Cartan class #3, occurring for 1 real form and for 3 dual real forms
atlas>
atlas> print_Cartan_info (H)
compact: 0, complex: 0, split: 2
canonical twisted involution: 2,1,2,1
twisted involution orbit size: 1; fiber size: 1; strong inv: 1
imaginary root system: empty
real root system: C2
complex factor: empty
atlas>
atlas> print_real_Weyl (G,H)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is trivial
A is trivial
W_ic is trivial
W^R is a Weyl group of type B2

generators for W^R:
1
2

A Cartan Subgroup isomorphic to \({\mathbb C}^\times \times {\mathbb C}^\times\) and Weylgroup of type B2. So the number of KGB orbits is 8/8=1:

atlas> set x=KGB(G,10)
Variable x: KGBElt (overriding previous instance, which had type KGBElt)
atlas>  x:=KGB(G,10)
Value: KGB element #10
atlas>

atlas> void: for w in W do prints(cross(w,x)) od
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
KGB element #10
atlas>

This concludes this deiscussion on \(K\backslash G/B\) orbits. In the next chapter we will discuss the representations associated to the intermediate Cartan subgroups. The parameter includes a discrete series of a Levi factor of a parabolic subgroup. So, to some extent it reduces to the case of discrete series.

The idea is to look at the cuspidal data of an arbitrary parameter which gives a Levi factor \(M\) and then applying what we learned about discrete series of M.