The "checkbasept" command checks a conjecture. We work in the context
of a fixed real form, and its KGB space X.
We may write a twisted involution w=s_1...s_p (see the help file for
the kgb command). We say this expression is involution-reduced if p is
minimal (see the help file for the blockd command).
Given an expression s_1...s_p we may apply it to a closed orbit x,
where conjugation corresponds to the cross action, and
left-multiplication to a Cayley transform.
Fix a fundamental parameter delta, i.e. the parameter of a large
fundamental series. The conjecture is that if w=s_1...s_p is
involution-reduced, then s_1...s_p.delta is independent of the choice of
reduced expression. By applying each twisted involution to delta
we obtain a "basepoint" in each fiber of the map X->twisted
involutions.
The checkbasept command prints out a list:
twisted involution tau, involution-reduced form w of tau, image of delta under w
where delta is the parameter of a large fundamental series.
The final line checks whether the conjecture is true for this block.
Example:
Here is the block of the trivial representation of Sp(4,R) (the block
command):
0( 0,6): 1 2 ( 6, *) ( 4, *) [i1,i1] 0
1( 1,6): 0 3 ( 6, *) ( 5, *) [i1,i1] 0
2( 2,6): 2 0 ( *, *) ( 4, *) [ic,i1] 0
3( 3,6): 3 1 ( *, *) ( 5, *) [ic,i1] 0
4( 4,4): 8 4 ( *, *) ( *, *) [C+,r1] 1 2
5( 5,4): 9 5 ( *, *) ( *, *) [C+,r1] 1 2
6( 6,5): 6 7 ( *, *) ( *, *) [r1,C+] 1 1
7( 7,2): 7 6 (10,11) ( *, *) [i2,C-] 2 212
8( 8,3): 4 9 ( *, *) (10, *) [C-,i1] 2 121
9( 9,3): 5 8 ( *, *) (10, *) [C-,i1] 2 121
10(10,0): 11 10 ( *, *) ( *, *) [r2,r1] 3 1212
11(10,1): 10 11 ( *, *) ( *, *) [r2,rn] 3 1212
Here is the output of checkbasept for this block:
(,,1)
(2,2,5)
(1,1,6)
(212,21,7)
(121,12,9)
(1212,121,10)
true