Last updated: October 8, 2005

Let G be a connected reductive complex algebraic group. The inner class of a real form of G is its image in the outer automorphism group of G. We look at inner classes up to conjugacy in Out(G); it may be shown that there are always finitely many possiblities.

Let Rad(G) be the identity component of the center of G. Then an inner class is entirely determined by the involutions it induces on the Dynkin diagram of G, and on Rad(G). If G is the direct product of Rad(G) and the derived group Der(G), and if Der(G) is simply connected, or adjoint, all pairs of involutions are allowed; this is no longer true in general (the simplest case is when G is PSL(2).SL(2); then the complex inner form (see below) is not allowed.)

The involution induced on the Dynkin diagram of G either fixes a component, or interchanges it with an isomorphic one. We require that the pairs of interchanged components correspond to consecutive entries in the Lie type (for instance, it is allowed to interchange the two A1's in B3.A1.A1, but not in A1.B3.A1.) Of course, it is always possible to lay out the group G in such a way that this condition is fulfilled. We now have a number of possibilities:

• A pair of interchanged components gives rise to a complex factor (i.e., a complex reductive algebraic group viewed as real);
• A fixed component on which the involution induces the identity will be labelled “c” (for compact), or also “e” (for equal rank). This corresponds to the fact that the inner class induced on that factor will be the one containing the compact real form, also called the equal rank inner class;
• For a fixed component of one of the types A_n, n >= 2, D_n, n odd, and E6, the non-identity involution of the Dynkin diagram is denoted “s”, because the inner class induced on that factor will be the one containing the split real form; this may also be denoted “u” (for unequal rank.) For the types A1, B_n, C_n, E7, E8, F4, G2, there is a unique inner class, which may be denoted either “c” or “s”;
• For a fixed component of type D_2m, the non-identity involution of the Dynkin diagram is denoted “u”;

As recalled here, any torus with involution can be split up into a direct product of factors on which the involution is either the identity (labelled “c”as above), inversion (labelled “s”), or the exchange of two isomorphic torus factors (labelled “C” for complex). By choosing the way we write Rad(G) appropriately, we may therefore assume that the involution is either the identity, the inversion, or the exchange of two consecutive isomorphic torus factors on the various factors of Rad(G).

The upshot is that without loss of generality we may represent our inner class by a sequence of symbols from the set {c,s,C,u}, one for each factor in the Lie type, except that a “C” symbol uses up two consecutive isomorphic factors, and with the restriction that "u" is allowed only when there is more than one inner class. If it turns out that a given sequence is not allowed for the specific covering group G that we are considering, the program will notice, and complain accordingly.

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