A vanishing theorem for modular symbols on locally symmetric spaces

Vanishing Theorem for Irreducible Symmetric Spaces of Noncompact Type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M 6= SO0(2, 2)/SO(2)×SO(2). Let π : E → M be any vector bundle, Then any E−valued L harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.

A Conjugacy Theorem for Symmetric Spaces

In this paper we prove that two Cartan subspaces of a semisimple symmetric pair (g, τ) are conjugate under G = Int(g) if and only if they are conjugate under (Gτ )0. Moreover we derive a double coset decomposition for G, which improves the results of Oshima and Matsuki.

A Paley–wiener Theorem for Distributions on Reductive Symmetric Spaces

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. We study Fourier transforms of compactly supported K-finite distributions on X and characterize the image of the space of such distributions.

A Fatou-type Theorem for Harmonic Functions on Symmetric Spaces

Holomorphic Torsion for Hermitian Locally Symmetric Spaces

Contents 1 Holomorphic torsion 5 2 The trace of the heat kernel 8 2.