Residual spectrum of $\mathrm{GL}_{2n}$ distinguished by $\mathrm{GL}_n \times \mathrm{GL}_n$

TL;DR

This paper develops a regularization approach to analyze linear periods of automorphic forms on $ ext{GL}_{2n}$, characterizing distinguished automorphic representations and establishing period vanishing results.

Contribution

It introduces a formula linking regularized periods to degenerate Whittaker functions and characterizes distinguished automorphic representations in the discrete spectrum.

Findings

Derived a formula for regularized periods in terms of degenerate Whittaker functions.

Characterized automorphic representations distinguished by $ ext{GL}_n imes ext{GL}_n$.

Proved vanishing of regularized periods when subgroup dimensions are unequal.

Abstract

Following the regularization method presented by Zydor, we study in this paper the regularized linear periods of square-integrable automormphic forms on $\mathrm{GL}_{2n}(\mathbb{A}_F)$, where $F$ is a number field and $\mathbb{A}_F$ its ring of adeles. We obtain a formula that expresses the regularized period of a noncuspidal, square-integrable automorphic form in terms of degenerate Whittaker functions in an inductive manner. As a consequence we characterize irreducible automorphic representations in the discrete spectrum of $\mathrm{GL}_{2n}(\mathbb{A})$ that are distinguished by $\mathrm{GL}_n(\mathbb{A}) \times \mathrm{GL}_n(\mathbb{A})$. We also show the vanishing of the regularized periods of square-integrable automorphic forms on $\mathrm{GL}_n(\mathbb{A})$ over $\mathrm{GL}_p(\mathbb{A}) \times \mathrm{GL}_q(\mathbb{A})$ when $p$ is not equal to $q$.