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Spectra of Compact Locally Symmetric Manifolds of Negative Curvature.

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TLDR
In this article, the authors studied the relationship of A to the geometry of X and determined the asymptotic growth of A as a subset of A(S) under the assumption that X is compact and defined the spectrum A of X as the set of those elements of A (S) for which one can find a nonzero eigenfunction defined on X.
Abstract
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 differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

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Journal ArticleDOI

Heat Kernel and Green Function Estimates on Noncompact Symmetric Spaces

TL;DR: In this paper, the authors obtained optimal upper and lower bounds for the heat kernel $h_t(x,y)$ (as well as asymptotics and estimates of its derivatives).
Journal Article

Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups

TL;DR: In this article, a Lie groupstion commerciale ou impression systématique is constitutive d'une infraction pénale, i.e., a copie ou an impression of a fichier do not conte-nir la présente mention de copyright.
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Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

TL;DR: Weak commensurabilty of arithmetic subgroups was introduced in this paper, which relates the notion of weak commensurality to the length equivalence and isospectrality of locally symmetric spaces.
References
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Book

Differential Geometry and Symmetric Spaces

TL;DR: In this article, the classification of symmetric spaces has been studied in the context of Lie groups and Lie algebras, and a list of notational conventions has been proposed.
Book

Lie groups, Lie algebras, and their representations

TL;DR: In this article, differentiable and analytic manifolds and Lie Groups and Lie Algebras have been studied in the context of structure theory and representation theory, and complex semisimple Lie Algebraic structures have been proposed.
Journal ArticleDOI

Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series

TL;DR: The trace-formula as discussed by the authors is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions) and is used in many of these works.
Book

Strong Rigidity of Locally Symmetric Spaces.

G. D. Mostow
TL;DR: In this paper, a metric definition of the Maximal Boundary is defined and a map of R-Rank 1 is presented. But this map does not cover all R-rank 1 spaces.
BookDOI

The Selberg trace formula for PSL (2, IR)

TL;DR: The selberg trace formula (version A) as mentioned in this paper is a trace formula for the Poincare series and the spectral decomposition of L2(? \H,?).