## Tables of Structure and Representation Theory

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 G_{2},
F_{4} and E_{8}) 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 E_{8}, and 603,032
irreducible represenations of them with infinitesimal character rho.
Of these 1+120+135=256 are discrete series representations, and
2^{8}=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