Real forms

Last updated: November 26, 2005

Let G be a connected complex reductive algebraic group. According to general principles in algebraic geometry, a real form of G is an antilinear involution of G; the set of fixed points of the involution is denoted G(R) and is a real reductive group, often non-connected. It turns out that there is a bijective correspondence between G-conjugacy classes of antilinear involutions of G, and of ordinary involutions of G as a complex algebraic group; to set this up, one chooses a compact real form of G, and conjugates both types of involutions to ones that commute with the chosen compact antilinear involution.

Therefore, in this program we always represent real forms through ordinary involutions of G.

To relate our data to perhaps more familiar objects, one should think of our involution θ as the complexification of a Cartan involution for G(R). In particular, the compact form of G corresponds to taking θ = Id, the identity involution. Similarly, one should think of the group K of θ-fixed points in G as the complexification of a maximal compact subgroup of G(R).

We consider two real forms to be equivalent if they are conjugate under G (a word of caution: the natural notion of isomorphism of real algebraic groups would translate to considering real forms up to conjugacy in the full automorphism group of G; however it is conjugacy under G that is the appropriate one for our purposes. An example where the difference is apparent is the case of type D_2m; a more elementary example is G = SL(2).SL(2), where SU(2).SL(2,R) and SL(2,R).SU(2) are obviously isomorphic groups, but not equivalent real forms in our sense.

In fact, what we have called real forms so far should be called weak real forms. There is a more subtle notion of strong real form, which should be thought of as some kind of “lifting” of the notion of a weak real form (indeed, when G is adjoint, the two notions coincide.) It turns out that the classifications of strong and weak real forms can be readily computed from the internal data that we keep in the program; in substance, the classifications amount to the computation of the orbits of an action of a subgroup of the Weyl group W of G on a small finite set (the precise statement about the classification of strong real forms may be found here.) The classification of weak real forms is printed out by the “showrealforms” command, and is used consistently in all instances where a choice of real form must be made. The various combinatorial types of strong real form packets are output by the “strongreal” command.


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