Eisenstein Series and the Trace Formula

In the first part of these notes we shall try to describe the main ideas in the theory. Let G be a reductive algebraic matrix group over Q. Then G(A) is the restricted direct product over all valuations v of the groups G(Qu). If v is finite, define Ku to be G(uJ if this latter group is a special maximal compact subgroup of G(Qu). This takes care of almost all v. For the remaining finite v, we let KO be any fixed special maximal compact subgroup of G(Qu). We also fix a minimal parabolic subgroup PO, defined over Q , and a Levi component Mo of Po. Let An be the maximal split torus in the center of My. Let Ko be a fixed maximal compact subgroup of G(R) whose Lie algebra is orthogonal to the Lie algebra of Ao(R) under the Killing form. Then K = FI "KO is a maximal compact subgroup of G(A). For most of these notes we shall deal only with standard parabolic subgroups; that is, parabolic subgroups P, defined over Q , which contain Po. Fix such a P. Let N p be the unipotent radical of P, and let M p be the unique Levi component of P which contains Mo. Then the split component, Ap, of the center of M p is contained in A,,. If X(Mp)n is the group of characters of M p defined over Q , define LIP = Hom((Xp)@, R) . Then if 111 = \{^mu lies in M(A) , we define a vector H,&) in rip by