Even rank type D

Last updated: October 8, 2005

Even rank type D is the type that causes by far the most headaches as far as the classification of inner classes and real forms is concerned. Type D4 is even a little bit more exceptional than the others; therefore we assume in this page that we are in type D_2m, with m > 2.

As for inner classes, this is the only irreducible Dynkin diagram possessing a non-trivial involution, for which the split form so(2m,2m) and the compact form so(4m) belong to the same inner class. The inner class corresponding to this non-trivial involution must be accessed through the “u” (unequal rank) symbol. The fundamental real form in this class is so(4m-1,1), and the quasisplit form is so(2m+1,2m-1).

These groups are also the only ones for which the simply connected group (i.e., Spin(4m)), has a center that is not cyclic, viz. Z/2.Z/2. This causes the printout of the center of the simply connected group to have more cyclic factors than there are factors in the Lie type, with the corresponding danger of confusion.

As for real forms, these groups are also the only simple Lie types for which there are distinct weak real forms corresponding to isomorphic real groups (in other words, there are involutions of G that are conjugate in the outer automorphism group but not in the inner one.) In fact, there is just one pair of weak real forms that exhibits this behaviour, viz. the two “versions” of so*(4m). In the program, we distinguish these as follows. There are three non-trivial elements in the center, which we denote [0,1], [1,0] and [1,1]. We arrange the correspondence in such a way that [1,1] is the element that is fixed by the non-trivial element of Out(G) (recall that we have assumed that we are not in type D4.) Then the quotient of Spin(4m) by the two-element subgroup generated by [1,1] is SO(4m), and the real forms corresponding to the two so*'s for this quotient are both connected. Now for each of the two other two-element subgroups in the center, one of the so*'s gives rise to a non-connected real form, and the other to a connected one. We denote so*(4m)[0,1] the one that gives rise to a non-connected real form in the quotient by the subgroup generated by [0,1], and so*(4m)[1,0] the other. (The computation of component groups is one of the capabilities of the program.)


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