In this work the author introduces a new method for establishing Rankin-Selberg type identities. In place of unfolding, he uses multiplicity one properties. Recall that if \(G\) is a real Lie group and \(H\) is a real Lie subgroup of \(G\) then the pair \((G,H)\) is called a strong Gelfand pair if for any representations \((\pi,V)\) of \(G\) and \((\sigma, W)\) of \(H\), the dimension of the space of \(H\)-equivariant maps of smooth vectors \(\text{Hom}_H(V,W)\) is at most one. The identities introduced here may be established when \({\mathcal G}\) is a real Lie group and \({\mathcal F}\subset {\mathcal H}_i\subset {\mathcal G}\), \(i=1,2\) are subgroups with the property that each of the pairs \(({\mathcal G},{\mathcal H}_i)\) and \(({\mathcal H}_i,{\mathcal F})\) are strong Gelfand pairs. The paper begins with a general description of how this set-up leads to Rankin-Selberg type identities when one computes the period of an automorphic form over the cycle associated to \({\mathcal F}\).
Next two cases are carried through in detail. Let \(G=\text{PGL}_2({\mathbb R})\), \(N\) be the standard unipotent subgroup of \(G\) and \(K=\text{PO}(2)\). In both cases, let \({\mathcal G}=G\times G\), \(\Delta\) be the standard diagonal embedding, and \({\mathcal H}_2=\Delta G\). For the first case, let \({\mathcal H}_1=N\times N\) and \({\mathcal F}=\Delta N\). Then the method introduced here leads to the classical Rankin-Selberg identity. This is explained, and the use of the identity with suitable test functions to estimate the standard (unipotent) Fourier coefficients of Maaßforms is also carried out. Strictly speaking, the uniqueness principle underlying the method is only almost satisfied for \(N\), but the theory of the constant term of the Eisenstein series is invoked to remedy this in the automorphic setting.
In the second case discussed in detail, \({\mathcal H}_1=K\times K\), \({\mathcal F}=\Delta K\). Using this, the author gives an anisotropic version of the classical Rankin-Selberg identity. As a consequence, he is able to give new bounds for the spherical Fourier coefficients of Maaßforms. (A similar bound for more general groups, but slightly weaker in this specific case, was proved using ergodic theory by A. Venkatesh [Sparse equidistribution problems, period bounds, and subconvexity, preprint, http://arxiv.org/abs/math/0506224].) Using a result of Waldspurger, this leads to a subconvexity bound in twisted aspect for an imaginary quadratic base change of a Maaßform. This is carried out in the setting of a co-compact discrete subgroup, so that \(\Gamma\backslash \mathbb H\) is a compact Riemann surface. In both settings, one needs to study certain integral transforms as well.
The method described here is very general. It has also been used by J. Bernstein and the author to give a subconvexity estimate for the triple \(L\)-function [ Subconvexity bounds for triple \(L\)-functions and representation theory. (Revised version to appear in Ann. Math. (2)), http://arxiv.org/abs/math/0608555].