arXiv:hep-th/0304130v1 15 Apr 2003 Spectral Theory of Automorphic Forms and Analysis of Invariant Differential Operators on SL 3 (Z) with Applications Sultan Catto † , Jonathan Huntley a , Nam-Jong Moh a and David Tepper a Physics Department University Center and The Graduate School The City University of New York 365 Fifth Avenue, New York, NY 10016-4309 and Center for Theoretical Physics The Rockefeller University New York, New York 10021-6399 We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl’s law and an analog of Selberg’s eigenvalue conjecture for SL3(Z) is given. We prove the following: Let H be the homogeneous space associated to the group P GL3(R). Let X =Γ\SL3(Z) and consider the first non-trivial eigenvalue λ1 of the Laplacian on L 2 (X). Using geometric considerations, we prove the inequality λ1 > 3π 2 /10 > 2.96088. Since the continuous spectrum is represented by the band [1, ∞), our bound on λ1 can be viewed as an analogue of Selberg’s eigenvalue conjecture for quotients of the hyperbolic half space. Brief comment on relevance of automorphic forms to applications in high energy physics is given. I. INTRODUCTION Automorphic forms are one of the central topics of analytic number theory: cornerstone of modern mathematics, they sit at the confluence of analysis, algebra, geometry and number theory with applications in physics of string theories, statisti- cal mechanics, infinite dimensional Lie algebras, cohomology and other areas of mathematical physics, as well as in com- puter networks. In recent past there has been many important developments in this field. Aside from Langland’s programme being the preoccupation of many mathematicians, remarkable progress has been made in the area of Abelian class field the- ory, structure theory of representation of groups over local fields (such as group representations and the reciprocal rela- tions between automorphic forms and Galois representations), Artin’s L-functions and their meromorphy, analytic proper- ties of L-functions of automorphic forms, multiplicity one theorems, complete reducibility of spaces cusp forms, finite dimensionality of spaces of automorphic forms, among oth- ers. After being initiated by Poincar´ e, Klein, Gauss, Jacobi, Riemann and Eisenstein, the automorphic forms (particularly modular forms) further flourished through outstanding works of Siegel, Hecke, Maass, Selberg and others. Sophistication of the formalism of string theories, black holes, M-theory, etc. that involves gauge theories, supersym- metry, general relativity, supergravity theories in higher di- mensions and quantum mechanics lead to interesting math- ematical physics questions and important new insights in pure mathematics. Particularly important are the string du- alities that relate different kind of string theories and have remarkable mathematical consequences. Mirror symmetry and algebraic curves in Calabi-Yau manifolds, string duality, where methods from the conformal field theory and from non- perturbative string theory leading to numerous predictions are some examples. Direct relations exist with quantum cohomol- ogy, symplectic geometry, Hodge theory, generalized Kac- Moody algebras and theory of automorphic forms (see, for example [1]-[8]). In this paper we will concentrate mainly on purely mathematical aspects of the theory, namely the spectral theory of automorphic forms and small and large eigenvalue questions. Some extensions of our formalism to complex Lie groups (in particular to L-groups of GL(4) and Spin(9)) that makes use of the exceptional group F 4 , connections to Cay- ley plane and exceptional Jordan algebra of dimension 27 will be given in another publication [9]. For a more in depth con- nections between automorphic forms and associated physics we refer to Greg Moore’s article which also has an extended bibliography [10]. Eigenfunctions of the Laplace operator on a Riemann man- ifold are of great interest for physicists working on a variety of problems. Especially the square-integrable eigenstates are of particular importance. Question of their behavior on high energy limits (with respect to the eigenvalues), their concen- tration onto specific manifolds or sets such as closed geodesics when being on distinguished energy levels, and the distribu- tion law for these levels are some of the problems that need to be worked out in detail. II. SPECTRAL THEORY OF AUTOMORPHIC FORMS, SMALL AND LARGE EIGENVALUE QUESTIONS AND APPLICATIONS Differential eigenvalue problems on manifolds have played an important role in physics and mathematics for many years. Many important questions in geometry and physics can be phrased in terms of these type of problems. When the mani- fold is actually a symmetric space one has both an increase in the number of subjects to which these problems are applica- ble and an increase in the number of techniques that become applicable. For example differential eigenvalue problems on symmetric spaces have applications in representation theory and number theory. Moreover techniques from these subjects become applicable. There is of course a close relationship between the spectral theory of automorphic forms and these subjects. We study various problems in the spectral theory of automorphic forms and these can all be recast in terms of dif- ferential eigenvalue problems on symmetric spaces. The type of problems we investigate can be roughly divided into two types: The first type is the study of the asymptotic distribu- tion of large eigenvalues of the Laplacian on fixed symmetric states where we look for results such as Weyl’s law and error terms in this expression. A related problem for large eigen- values is the multiplicity problem. In dealing with this issue one tries to obtain good upper bounds for the dimension of the space of eigenfunctions of the Laplacian with eigenvalue λ. Another related problem is the comparison problem, where one attempts to find a finite amount of data that will imply that two automorphic forms (eigenfunctions) are equal. We will