On the unramified spherical automorphic spectrum

For a split connected reductive group $G$ defined over a number field $F$, we compute the part of the spherical automorphic spectrum which is supported by the cuspidal data containing $(T,1)$, where $T$ is a maximal split torus and $1$ is the trivial automorphic character. The proof uses the residue distributions which were introduced by the third author (in joint work with G. Heckman) in the study of graded affine Hecke algebras, and a result by M. Reeder on the weight spaces of the (anti)spherical discrete series representations of affine Hecke algebras. Note that both these ingredients are of a purely local nature. For many special cases of reductive groups $G$ similar results have been established by various authors. The main feature of the present proof is the fact that it is uniform and general.