The "KGB" command is an enhanced version of the "kgb" command (see the
help file for kgb). KGB gives the following information, in addition
to that provided by kgb.
Each row of the output of kgb or KGB corresponds to an orbit of K on
G/B. This corresponds to an element x of the extended group.
In the output of kgb the third column gives the number of the
corresponding Cartan subgroup. In the output of KGB this is moved to
the second to last column.
Every kgb element maps to a twisted involution in the Weyl group.
This is an element of w satisfying w*delta(w)=1 where delta is the basic
distinguished involution for the inner class. The element w is given
by the last entry in the row. The Cartan subgroups are parametrized by
conjugacy classes of such involutions. Each conjugacy class contains a
unique "canonical" involution. The elements of kgb for which the
corresponding involution is canonical are denoted by a "#" preceding
the Cartan class number.
The elements x lying over the same twisted involution are parametrized by a
product of m copies of Z/2Z for some m (see the strongreal command). This
information is given by a parenthesized vector of 0s and 1s, of size the rank
of G, given before the indication of the Cartan class and involution.
Example: KGB for Sp(4,R)
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):
Base grading: [11].
0: 0 [n,n] 1 2 4 5 (0,0)#0 e
1: 0 [n,n] 0 3 4 6 (1,0)#0 e
2: 0 [c,n] 2 0 * 5 (0,1)#0 e
3: 0 [c,n] 3 1 * 6 (1,1)#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
Since Sp(4,R) contains a compact Cartan subgroup \delta acts trivially
on G and W, and twisted involutions are ordinary involutions.
Elements 0-3 all correspond to the compact Cartan subgroup (number 0),
with trivial involution. This involution is the only one for the Cartan
and is therefore canonical. The fiber at this Cartan has 4 elements (as
reported by the cartan command), all of which belong to our real form
sp(4,R) (as reported by strongreal), so the KGB elements 0-3 are
distinguished by the values in Z/2Z^2.
Elements 4,9 correspond to Cartan number 1; the two involutions 1^e and
2x1^e occur, the latter is canonical. The fiber at this Cartan has only
one element, so the fiber parts (both (0,0)) do not discriminate here.
Elements 5-8 correspond to Cartan number 2. The involutions are 2^e and
1x2^e (canonical), and the fiber has size 2, all told 4 elements. Note
that the set of values for the fiber part is not the same for the two
involutions.
Finally there is only one element x=10 on the split Cartan number 3.
The precise meaning of the vector of 0s and 1s of the fiber part depends
on the "base grading" used for the entire KGB set, and given at the
beginning out the output of the KGB command (here [11] indicates a
non-compact base grading on both simple roots). Both the actual gradings
at each KGB element (third column) and the cross actions and Cayley
transforms (fourth through seventh column) have been computed with aid
of the base grading and the fiber parts of the individual KGB elements.