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arXiv · 2601.02542

The fine spectral expansion of the Rankin-Selberg period

Paul Boisseau
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Abstract

We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration \`a la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber.

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Paul Boisseau. 2026-01-05. The fine spectral expansion of the Rankin-Selberg period. https://arxiv.org/abs/2601.02542

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