The "blockstabilizer" command describes the structure of the
stabilizer in the Weyl group of a block element x. This is a subgroup W_x
of the real Weyl group W(G,H_x) for the corresponding Cartan subgroup
H_x (see the help file for "realweyl"). The subgroup depends up to
conjugacy only on the block and the Cartan subgroup; the software
therefore asks only for these two pieces of information. (As usual the
block is specified by a real form of the dual group, and the Cartan
subgroup by a number explained by the "cartan" command.)
The structure of W_x, like that of W(G,H_x), is described in Vogan,
Irreducible Characters of Semisimple Lie Groups IV, Duke Math J. 49,
no. 4, pp. 943--1073. Recall that W(G,H_x) has the structure
W^C.((A_i.W_ic) x W^R)
explained in the help for "realweyl." The subgroup W_x has the
structure
W^C.((A_i.W_ic) x (A_r.W_rc)).
Here W_rc is the Weyl group of the real roots not satisfying the
parity condition; these correspond to the even coroots in a
Z/2Z-grading of the real roots. The group A_r is an elementary abelian
2-group normalizing W_rc. The structure of A_r.W_rc is therefore
entirely analogous to that of A_i.W_ic. In fact A_r.W_rc may be
identified with the compact imaginary Weyl group (A^v_i.W^v_ic) inside
W^v_i in the Weyl group of the dual Cartan for the dual block in G^v.
Just as for realweyl, only the 2-groups A_i and A_r are subtle and
depend on the group (as opposed to the Lie algebra).
Note that the number of block elements attached to the Cartan H_x is
equal to the index of W_x in W.