Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms

Reznikov, Andre

doi:10.1090/S0894-0347-07-00581-4

Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms
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by Andre Reznikov

J. Amer. Math. Soc. 21 (2008), 439-477

DOI: https://doi.org/10.1090/S0894-0347-07-00581-4

Published electronically: October 4, 2007

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Abstract:

We use the uniqueness of various invariant functionals on irreducible unitary representations of $PGL_2(\mathbb {R})$ in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain $L$-functions. Our main tool is the notion of a Gelfand pair from representation theory.

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