Andre Reznikov

Analytic continuation of representations and estimates of automorphic forms

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (�,G,V ) a continuous representation of G in a topological vector space V. A vector v ∈ V is called analytic if the functionv : g 7→ �(g)v is a real analytic function on G with values in V. This means that there exists a neighborhood U of G in its complexification GC such thatv extends to a holomorphic function on U. In other words, for each element g ∈ U we can unambiguously define the vector �(g)v asv(g), i.e., we can extend the action of G to a somewhat larger set. In this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates. Unless otherwise stated, G = SL(2, R), so GC = SL(2, C). We consider a typical representation ofG, i.e., a representation of the principal series. Namely, fix� ∈ C and consider the space Dof smooth homogeneous functions of degree � − 1 on R 2 \ 0, i.e., D� = {� ∈ C ∞ (R 2 \ 0) : �(ax,ay) = |a| �−1 �(x,y)}; we

Nodal Domains of Maass Forms I

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L2-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L∞-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

Sobolev norms of automorphic functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the representation theory (eventually, of adelic groups, see [7, 8, 9]). Later, Ol’shanski gave a more transparent proof of it (see [14]). However, in the subsequent rapid development of the theory of automorphic functions, Frobenius reciprocity was barely noticeable. We believe that this is due to the incompleteness of the above-mentioned results. In this paper, we prove a general theorem (see Theorem 1.1), which we view as a quantitative version of Frobenius reciprocity. We then illustrate it by looking into the example of SL(2,R). We think that these methods will play a more prominent role in the theory of automorphic functions.

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

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