Cartan subgroups

Last updated: October 15, 2005

Let G be a connected complex reductive algebraic group, with a fixed inner class of real forms, and fix a real form in the class, represented by an involution θ. Let K be the fixed-point group for θ in G. The “cartan” command will output representatives of the various K-conjugacy classes of θ-stable Cartan subgroups in G. (These are in natural bijection with the G(R)-conjugacy classes of Cartan subgroups of G(R)).

It turns out that there is a rather nice connection between the classifications of Cartan subgroups of the various real forms of G belonging to the fixed inner class. For the quasisplit form, the Cartans are in (1,1) correspondence with (twisted) W-conjugacy classes of twisted involutions in W (a twisted involution is an element w in W satisfying w.δ(w)=1, where δ is deduced from the involution of the Dynkin diagram defined by the inner class.) This set TwConj(W,δ) carries a natural poset structure, with a minimal element cmin which is the fundamental Cartan. For any other real form θ, the Cartans of (G,θ) may be identified with the interval [cmin,cmax] in TwConj(W,δ), where cmax corresponds to the most split Cartan in (G,θ).

For each conjugacy class, the program will output the type of the Cartan subgroup as a real algebraic torus; note that this is relatively subtle, as it depends not only on the Lie type of G but also on the actual covering group chosen: a complex torus factor might become a compact times split one, or conversely (try for instance G = SL(2).SL(2), and G1 = G/D, where D is the diagonal subgroup in the center of G, both for the complex real form (which is an inner class all by itself.) For G(R), you get a complex torus, which makes sense because the group is just SL(2,C); whereas for G1(R), which is SO(3,1), you get a compact times split torus. For G2 = G/Z(G), which has G2(R)=PSL(2,C), you again get a complex torus.)

Then the program will output data that depend only on the restriction of θ to the Cartan (and therefore are the same for all the real forms sharing this Cartan): the real and imaginary root systems, and what I call, for lack of a better name, the "complex factor", which is the complex root system defined in [1], orthogonal to both the half-sums of positive real and of positive imaginary roots. From these data, as explained in [1], one can easily deduce the group W(θ) of θ-fixed elements in W.

Finally, the program will output the classification of the weak real forms of G for which this Cartan is defined. This amounts to the classification of the orbits of the Weyl group of the imaginary root system, in a set that carries a simply transitive action of the component group of the torus dual to the corresponding Cartan in the adjoint group of G (notice that the Cartan classifications for G and its adjoint group are the same, even when G is reductive.) To obtain the classification of strong real forms for this Cartan, (or more precisely, the various combinatorial types of packets of strong real forms, as explained here), use the “strongreal” command.

[1] David A. Vogan, Jr., Irreducible characters of semisimple Lie groups IV. Character-multiplicity duality, Duke Math. J. 49 (1982), no. 4, pp. 943--1073.

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