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

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 $q$.

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