Spectra of compact locally symmetric manifolds of negative curvature

  title={Spectra of compact locally symmetric manifolds of negative curvature},
  author={Johannes Jisse Duistermaat and J. A. C. Kolk and V. S. Varadarajan},
  journal={Inventiones mathematicae},
Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \ S . Then the members of 5~ may be regarded as… 
On the distribution of the principal series in ²(Γ\)
Let G be a semisimple Lie group of split rank one with finite center. If T C G is a discrete cocompact subgroup, then L2(V\G) = 2ue6(C)nr(<,>) ' "■ For fixed o G &(M), let P(a) denote the classes of
Absence of principal eigenvalues for higher rank locally symmetric spaces
. Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no L 2 -eigenvalues ≥ 1 / 4. In this article
We report progress on the equidistribution problem of automorphic forms on locally symmetric spaces. First, generalizing work of Zelditch-Wolpert we construct a representation theoretic analog of the
On the Poisson relation for compact Lie groups
Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: Can the length spectrum of a closed Riemannian manifold be recovered from its Laplace
Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One
Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime
Asymptotics of Elementary Spherical Functions
Publisher Summary The most fruitful approach to the analysis of symmetric spaces is via the group theory. A negatively curved symmetric space S is a space on which a noncompact connected semisimple
On the Logarithmic Derivative of Zeta Functions for Compact, Odd-dimensional Hyperbolic Spaces
Abstract: In this paper we investigate the Selberg zeta functions and the Ruelle zeta functions associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. Our
On Quantum Unique Ergodicity for Locally Symmetric Spaces
Abstract.We construct an equivariant microlocal lift for locally symmetric spaces. In other words, we demonstrate how to lift, in a semi-canonical fashion, limits of eigenfunction measures on locally
On the Length Spectrum for Compact, Odd-dimensional, Real Hyperbolic Spaces
Abstract: We derive a prime geodesic theorem for compact, odd-dimensional, real hyperbolic spaces. The obtained result corresponds to the best known result obtained in the compact, even-dimensional
Higher rank quantum-classical correspondence
For a compact Riemannian locally symmetric space Γ\G/K of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an


Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces
Let G be a connected noncompact semisimple Lie group with finite centre C, and let K be a maximal compact subgroup of G. Let F be a discrete subgroup of G such that the quotient F\G is compact. F
Compact Clifford-Klein forms of symmetric spaces
Spectra of discrete uniform subgroups of semisimple Lie groups
Let G be a connected, semisimple Lie group with finite center and without compact factors, F a discrete uniform subgroup of G, U the unitary representation of G on L2(F\G) and [U] the equivalence
On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces
A formula is derived for the Minakshisundaram-Pleijel zeta function in the half-plane Re i < 0. Let S be a compact Riemann surface, which we will regard as the quotient of the upper half-plane H by a
Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds
Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (x i ), i = 1, 2, … Ν, the line-element dr is given by
On the Selberg trace formula in the case of compact quotient
It is a standard fact (see §2) that 7rr(4>) is of trace class. In particular, 7Tr(4>) is completely continuous for 4>eC7(G). This implies that L(r\G) decomposes into an orthogonal direct sum of
The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator
The methods depend on the fact that the Riemann hypothesis for the Selberg zeta-function is almost true, in the sense that any possible exceptional zeros are all located in the real segment (0, 1). I
An asymptotic formula of Gelfand and Gangolli for the spectrum of $G\backslash G$
In [6], Gelfand outlined a proof of an asymptotic formula for the distribution of multiplicities of spherical principal series in U(Γ\G), where G is a connected semi-simple Lie group with finite
On the Plancherel Formula and the Paley-Wiener Theorem for Spherical Functions on Semisimple Lie Groups
One of the difficult points in the proof of Harish-Chandra's Plancherel formula for spherical functions on a semisimple Lie group is to show that an appropriate inversion formula exists for the
Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series
In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations