Periods of automorphic forms over reductive subgroups

We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G^{\mathrm{\prime }}\subset G$. As a consequence we obtain an explicit $G^{\mathrm{\prime }}(\mathbb{A})$-invariant functional on the space of automorphic forms on $G$ whose exponents avoid certain prescribed hyperplanes. We also provide a necessary and sufficient condition for convergence of period integrals of automorphic forms in terms of their exponents.