For more information on the structure theory discussed here see Parameters for Real Groups and Algorithms for Structure Theory. The information here was produced by Fokko du Cloux using the atlas software package.
Each file gives information about all of the real forms in a given inner class. Listed are conjugacy classes of Cartan subgroups of the quasisplit form of G. Each Cartan is described as a real torus, and its real and imaginary root systems are given.
The other two numbers that are given for each Cartan relate to the way representations will be parametrized in the program. To each Cartan there is associated a conjugacy class of twisted involutions for the complex Weyl group. The parameter space for the representations corresponding to this Cartan is naturally fibered over this orbit; so knowing the size of the orbit and the size of the fiber we can compute their number.
Adding up the numbers for the various Cartans, we get the total number of parameters for this inner class. Roughly: this is the number of irreducible representations of all real forms of G, with regular integral infinitesimal character, modulo translation. This is imprecise for two reasons: we need to talk about strong real forms here, and the infinitesimal character statement requires some explanation.
If G is both simply connected and adjoint (i.e. for G2, F4 and E8) the statement is simple: the number at the end is the number of irreducible representations with infinitesimal character rho, of all of the real forms of G.
For example, there are 3 real forms of E8, and 603,032 irreducible represenations of them with infinitesimal character rho. Of these 1+120+135=256 are discrete series representations, and 28=256 are principal series representations of the split group.
Note that 603,032 is a remarkably small number, when compared to the size of the complex Weyl group, which is 696,729,600. This fact makes us hopeful that it will be possible to carry out substantial computations even for the split real form of E8.
Tables of structure theory for the exceptional groups
Tables of representation theory - currently includes A1, G2, F4, E6, some results for E7 and E8. We're currently adding more results here.
Computing the Unitary Dual Atlas Home Page