The "strongreal" command prints information about the strong real forms
for the current inner class of a reductive group G. It gives the closest
possible approximation to listing all of them for a given root datum
involution, suppressing only the trivial repetition due to a translation
action of the (possibly infinite) center of G
Recall that G is endowed with a Borel subgroup B containing a Cartan H,
a pinning, and an involutive automorphism delta preserving the pinning.
The extended group G^Gamma is the semi-direct product of G with Gamma =
{1, delta}. A _strong involution_ is by definition an H-conjugacy class
x of elements xi in the non-identity coset of G^Gamma, normalizing H,
such that xi^2 is central in G. The conjugation action of xi defines an
involutive automorphism theta_xi of G, which is the Cartan involution of
the corresponding "weak real form."
Two strong involutions are _equivalent_ if they are conjugate by G.
A _strong real form_ is an equivalence class of strong involutions.
Each strong involution defines an involution tau_x of the root datum;
this is the restriction to H of a Cartan involution theta_xi. There are
only finitely many of these involutions tau that can be realized using
our fixed G^Gamma. The Weyl group W acts by conjugation on the set of
involutions tau. Each conjugacy class of Cartan subgroups for the real
form corresponds to a W-conjugacy class of involutions. (If the real
form is quasisplit, this correspondence is surjective.)
The command "strongreal" begins by asking for a Cartan subgroup in the
inner class; that is, for a W-conjugacy class of root datum involutions.
The software essentially fixes a representative involution tau_0. The
output of the command is a list with one entry for each strong
involution x such that tau_x is equal to tau_0, or more precisely for
each coset for Z(G) of such strong involutions, so as to suppress
possibly infinite repetition as mentioned above.
The output is organized first by "square class". If x is a strong
involution, then x^2 = c is an element of Z(G)^delta (fixed points of
the basic automorphism delta on the center). If x is multiplied by a
central element z, then its square is multiplied by z.delta(z). Write
{z delta(z) | z in Z(G)} = (1 + delta)Z.
The _square class of x_ is by definition the image of x^2 in
Z^delta/(1+delta)Z. It is already determined by the (weak) real form
defined by x.
The quotient group Z^delta/(1+delta)Z is finite; in fact it is a
product of copies of Z/2Z. Not every element of this quotient group
can arise from a strong involution over the given tau_0; those which
do form a subgroup. The software computes this subgroup in abstract
form, and lists its elements as #0, #1,... #(2^r - 1). When printing
each class, the software indicates a representative element z_0 of
the class as an image by exp(2i\pi.) of an element of X_* \tensor \Q.
For each square class, the software then finds all the strong
involutions x lying over the fixed tau_0, with square equal to z_0.
It turns out that these x are parameterized by the "fiber group for
tau_0", which is a product of m copies of Z/2Z for some m (which is
bounded by the rank of the +1 eigenspace of tau_0). The software
numbers these 2^m strong involutions as 0, 1, 2,... 2^m -1.
Each line of the output shows all the strong involutions belonging to a
fixed strong real form, in square brackets. The line is labelled at the
beginning by the number of the corresponding weak real form of G (see
the "realform" command) and at the end (for convenience) by the number
of strong involutions in the equivalence class, in parentheses.
Example: strong real forms of Spin(4,4):
empty: strongreal
Lie type: D4 sc s
choose Cartan class (one of 0,1,2,3,4,5,6): 0
Name an output file (return for stdout, ? to abandon):
there are 4 real form classes:
class #0, possible square: exp(2i\pi([0,0,0,0]/2))
real form #4: [0,1,2,4,5,7,8,9,11,12,13,14] (12)
real form #0: [3] (1)
real form #0: [6] (1)
real form #0: [10] (1)
real form #0: [15] (1)
class #1, possible square: exp(2i\pi([1,0,0,1]/2))
real form #2: [0,1,2,6,8,9,11,15] (8)
real form #2: [3,4,5,7,10,12,13,14] (8)
class #2, possible square: exp(2i\pi([1,0,1,0]/2))
real form #3: [0,2,3,5,6,7,8,13] (8)
real form #3: [1,4,9,10,11,12,14,15] (8)
class #3, possible square: exp(2i\pi([0,0,1,1]/2))
real form #1: [0,2,3,4,8,12,14,15] (8)
real form #1: [1,5,6,7,9,10,11,13] (8)
The first line says that there are four possible values for the
square class in Z^delta/(1+delta)Z. (In fact Z(G) is Z/2Z x Z/2Z,
and delta acts trivially; so Z^delta = Z/2 x Z/2, and (1+delta)Z is
trivial.) The class #0 corresponds to the (weak) real forms numbered
4 and 0 (which are so(4,4) and so(8) respectively). The involution
tau_0 turns out to be trivial (since the Cartan is compact; this is
not displayed anywhere here). We are interested in extending the
trivial automorphism of the root datum to G.
The fiber group has order 16, so each class has 16 strong
involutions numbered 0 to 15. (The numberings within different
classes have nothing simple to do with each other.) In class #0, the
first line represents the 12 strong involutions in the unique
equivalence class (strong real form) defining the split real form
so(4,4). The next four lines represent the four distinct strong
involutions (their strong involutions are non-conjugate in G), each
defining the trivial automorphism of G, the compact real form so(8).
In class #1, we find two different strong real forms, each
representing the real form so*(8)[0,1], and each with eight
different strong involutions.
Classes #2 and #3 behave in the same way. In fact these three
classes, and the corresponding real forms, are permuted by the
diagram automorphisms of D4.