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

Abstract

In this paper we study periods of automorphic functions. We present a new method which allows one to obtain non-trivial spectral identities for weighted sums of certain periods of automorphic functions. These identities are modelled on the classical identity of R. Rankin [Ra] and A. Selberg [Se]. We recall that the RankinSelberg identity relates the weighted sum of Fourier coefficients of a cusp form φ to the weighted integral of the inner product of φ with the Eisenstein series (e.g., formula (1.7) below). We show how to deduce the classical Rankin-Selberg identity and similar new identities from the uniqueness principle in representation theory (also known under the following names: the multiplicity one property, Gelfand pair). The uniqueness principle is a powerful tool in representation theory; it plays an important role in the theory of automorphic functions. We associate a non-trivial spectral identity to certain pairs of different triples of Gelfand subgroups. Namely, we associate a spectral identity (see formula (1.4) below) with two triples F ⊂ H1 ⊂ G and F ⊂ H2 ⊂ G of subgroups in a group G such that pairs (G,Hi) and (Hi,F) for i = 1, 2 are strong Gelfand pairs having the same subgroup F in the intersection (for the notion of Gelfand pair that we use, see Section 1.1.3). We call such a collection (G,H1,H2,F) a strong Gelfand formation. In the Introduction we explain our general idea and describe how to implement it in order to reprove the classical Rankin-Selberg formula. We also obtain a new anisotropic analog of the Rankin-Selberg formula. We present then an analytical application of these spectral identities towards non-trivial bounds for various Fourier coefficients of cusp forms. The novelty of our results lies mainly in the method, as we do not rely on the well-known technique of Rankin and Selberg