The "kgp" command outputs a description of the K orbits on G/P,
where P is a parabolic subgroup containing the fixed Borel
subgroup B. Here K is the complexification of a maximal compact
subgroup for a particular (weak) real form. The software will
prompt the user for this if it has not already been chosen.
Once a real form is specified, the user is prompted to enter a
subset of the simple roots corresponding to the desired parabolic.
The input should be a list of integers in [1,rank], separated
by any whitespace or non-numeric character(s). Numbers outside this
range are ignored. An empty list corresponds to the case
B=P and regenerates K\G/B as a partially ordered set.
The output is as follows:
Note first that each K orbit on G/P is a union of K orbits on G/B
and the output is therefore in terms of the latter set. See the
command 'kgb' for more information about K orbits on G/B.
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). The orbit number is immediately followed by a 'starred
number' (in brackets) representing the unique K\G/B of maximal dimension
it contains.
The next two columns represent the (normalized) dimension of the orbit
followed by the number of K\G/B elements in the fiber of the projection
K\G/B --> K\G/P
over this orbit. The final column contains a comma separated list of the
individual K\G/B orbits contained in this fiber. As mentioned, see the
'kgb' command to make sense of these numbers.
For example:
empty: type
Lie type: B2 sc s
main: realform
(weak) real forms are:
0: so(5)
1: so(4,1)
2: so(3,2)
enter your choice: 2
real: kgp
enter simple roots (1-2): 1
kgp size for roots {1}: 4
Name an output file (return for stdout, ? to abandon):
0:[ 4*] 1 3 {0,1,4}
1:[ 5*] 1 3 {2,3,5}
2:[ 7*] 2 2 {6,7}
3:[10*] 3 3 {8,9,10}
Here the quasi-split form of type B2 is chosen, with P corresponding
to the root subset consisting of the unique long simple root. There
are 4 orbits of K on G/P. Orbits 0 and 1 each have (normalized)
dimension 1 and respectively contain K\G/B orbits 4 and 5 as dense
subsets. Orbits 0, 1, and 3 are each a union of 3 distinct K\G/B
orbits, whereas orbit 2 is a union of only two.