The "kgb" command outputs the information describing the K orbits on
G/B. This is for a particular (weak) real form, and the software will
prompt the user to choose this if it has not been chosen yet.
Each row of the output corresponds to a single orbit, and the first
column is a numbering of the orbits from 0 to n-1 (with n the number
of orbits).
Next there are two columns giving the length and the Cartan class to
which this orbit belongs (the length of orbit O is dim(O)-dim(O_min)
where O_min is an orbit of minimal dimension; this minimal dimension
is the number of positive roots for the reductive group K).
Follows a sequence of r letters in brackets (where r is the number of
simple roots) giving the types of the simple roots: 'c' (compact imaginary),
'n' (non-compact imaginary), 'C' (complex) or 'r' (real).
The next r columns give the cross actions of the simple roots on this orbit,
by specifying the number of the orbit reached from the current one by the
cross action in question.
The next r columns give the action of Cayley transforms by simple roots that
are imaginary and non-compact (in other cases '*' is printed).
There is a map p:X->W^Gamma, the latter being the twisted Weyl
group. We have fixed a "fundamental" element delta in W^Gamma-W, and
this element acts on W (permuting the simple reflections that generate
W). (If G contains a compact Cartan subgroup this action is
trivial). Then tau=p(x)=w*delta. We say w is a "twisted involution":
w*delta(w)=1. The identity is denoted e; the corresponding
involution is tau=delta. The last column gives the element w as
obtained from this initial involution by a series of multiplications
(by a commuting generator, so that the result is again an involution)
and conjugations.
For example:
empty: kgb
Lie type: C2 sc s
(weak) real forms are:
0: sp(2)
1: sp(1,1)
2: sp(4,R)
enter your choice: 2
kgbsize: 11
Name an output file (return for stdout, ? to abandon):
0: 0 0 [n,n] 1 2 4 5 e
1: 0 0 [n,n] 0 3 4 6 e
2: 0 0 [c,n] 2 0 * 5 e
3: 0 0 [c,n] 3 1 * 6 e
4: 1 1 [r,C] 4 9 * * 1^e
5: 1 2 [C,r] 7 5 * * 2^e
6: 1 2 [C,r] 8 6 * * 2^e
7: 2 2 [C,n] 5 8 * 10 1x2^e
8: 2 2 [C,n] 6 7 * 10 1x2^e
9: 2 1 [n,C] 9 4 10 * 2x1^e
10: 3 3 [r,r] 10 10 * * 1^2x1^e
These are the 11 orbits of K on G/B for G=Sp(4,R). In this case the
outer automorphism is trivial, twisted involutions correspond to
involutions in W, and delta=1. There are 6 involutions in W: the
identity e; simple reflections 1, 2 (printed as 1^e, 2^e, where the
caret stands for multiplication, as occurs in Cayley transforms);
the conjugates of each simple reflection by the other, printed as
2x1^e and 1x2^e (the cross denotes conjugation); and the long element
printed as 1^2x1^e. Of these 2^e and 1x2^e are (twisted) conjugate
(Cartan class #2) as are 1^1 and 2x1^e (Cartan class #1); the other
two form a conjugacy class by themselves (Cartan classes #0 and #3).
There are 4 minimal orbits, which are projective lines CP^1 in this case.
The cross action permutes these 4 lines among themselves, while Cayley
transforms take them to orbits 4-6. The final (open) orbit alone
maps to the long element of W.