Induced conjugacy classes, prehomogeneous varieties, and canonical parabolic subgroups

We prove the properties of induced conjugacy classes, without using the original proof by Lusztig and Spaltenstein in the unipotent case, by adapting Borho’s simpler arguments for induced adjoint orbits. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of canonical parabolic subgroups attached to elements of reductive groups in the sense of Jacobson-Morozov. These results are prerequisites for making the geometric expansion of the Arthur-Selberg trace formula more explicit. Mathematics Subject Classification: 20G15, 14L30 This paper contains prerequisites for an explicit description of certain coefficients that appear on the geometric side of Arthur’s trace formula for a reductive group G (see section 19 of [3] for an introduction). Those coefficients grew out of an invariance argument that did not allow their determination. In previous work on groups of low rank ([8], [11]), the coefficients in question had been related to certain prehomogeneous zeta integrals. In a forthcoming paper, we aim to generalise that approach to groups of general rank. Before entering the analytical argument, one has to identify, in the structure of the group G, the prehomogeneous vector spaces supporting those zeta integrals. We will see that they are closely related to the canonical parabolic subgroups Q of the elements of G. Actually, in the case of non-unipotent elements, prehomogeneous affine spaces appear as well. More general prehomogeneous varieties enter the stage in still another role, which can be explained as follows.