Strong real forms

Last updated: October 15, 2005

Let G be a connected reductive complex algebraic group. A pinning of G is a triple (T,B,{Xα}α∈Π), where T is a maximal torus in G, B is a Borel containing T, Π is the set of simple roots of (G,T) with respect to B, and for each α∈Π, Xα is a root vector for α in the Lie algebra of G; we fix such a pinning ℘ (containing our already chosen T and B) in all that follows. It is well-known that the adjoint group Int(G)=G/Z acts simply transitively on the set of pinnings. It follows that the exact sequence

1 → Int(G) → Aut(G) → Out(G) → 1

splits by identifying Out(G) with the stabilizer of ℘ in Aut(G). In particular, each inner class of real forms of G has a canonical representative θf fixing ℘; we will say that θf is the fundamental involution in the inner class.

Let Γ=Z/2, and write Γ={1,δ}. Then we may form the semidirect product

GΓ = G×|{1,δ}

where the conjugation action of δ on G is through θf. This is an analogue in our situation of the Langlands group, together with a specific choice of splitting; we will be mostly concerned with the non-identity component G.δ in GΓ.

A strong involution in GΓ is an element x in G.δ such that x2∈Z(G); a strong real form of G is a G-conjugacy class of strong involutions. Each strong involution x defines an ordinary involution θx of G by setting θx = int(x); it is clear that in this way we obtain exactly all involutions in the inner class defined by θf. By passing to conjugacy classes, to each strong real form corresponds a real form as defined here. When G is adjoint, the notions of real form (in the chosen inner class) and strong real form coincide.

Let Wim be the Weyl group of the imaginary root system for the fundamental involution θf. As mentioned here, the classification of strong real forms reduces essentially to an orbit computation for an action of Wim. Precisely, the strong involutions of G may be partitioned according to the values of x2=z ∈ Z(G); for a fixed value of z, there is a simply transitive action of the component group of the torus dual to the fundamental torus of G, on the set of strong involutions with square z that induce on T the same involution as θf. Then the orbits of Wim on that set of involutions classify the strong real forms of G with square z. Moreover, multiplication by elements of Z will obviously induce isomorphisms among “packets” of strong real forms; so packets that have the same image in the adjoint group give rise to the same orbit problem, and it is really enough to solve it once for each of the “packet images” in the adjoint group. This is what the “strongreal” command does; the classes of real forms mentioned in its output are the packet images. Then all that remains to be done to complete the classification is to count the number of packets, which may be in fact be infinite for some reductive groups (precisely, it is infinite if and only if the radical of G contains a compact factor.)

By a similar procedure, one may also classify the strong real forms for which any given conjugacy class of Cartan subgroups is defined. The argument to the “strongreal” command specifies the Cartan class.


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