  For an unramified connected reductive group $G$ defined over a number field
$F$, consider the part of the spherical automorphic spectrum with cuspidal
support $[T,\mathcal{O}(\chi)]$, where $T$ is a maximal torus and $\chi$ is an
unramified automorphic character. We define a normalization of the Eisenstein
series and we give the precise spectral decomposition of the closure of the
subspace spanned by the normalized pseudo-Eiseinstein series. The proof uses
residue distributions which were introduced by the third author (in joint work
with G. Heckman) in the study of graded affine Hecke algebras, which is an
ingredient of a purely local nature.
  In the case when $G$ is split and $\chi$ is the trivial character, we show
that the normalized spectrum is in fact the whole spherical automorphic
spectrum. The necessary argument to conclude the result in the split case are
based on combinatorial results proved in [DMHO].
