# Periods of automorphic forms over reductive subgroups

## TL;DR

This paper introduces a regularization method for period integrals of automorphic forms over reductive subgroups, providing explicit invariant functionals and convergence criteria based on exponents.

## Contribution

It offers a new regularization procedure for period integrals over arbitrary reductive subgroups, along with explicit invariant functionals and convergence conditions.

## Key findings

- Regularization procedure for period integrals over reductive subgroups
- Explicit $G'(A)$-invariant functional on automorphic forms
- Necessary and sufficient conditions for convergence based on exponents

## Abstract

We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-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.