Gaussian Weighted Unreduced L2 Cohomology of Locally Symmetric Spaces

Let (M, g) be a complete, noncompact Riemannian manifold of finite volume. For w : M → (0,∞) a weighting function, the w weighted unreduced L2 cohomology is defined as the usual unreduced L2 cohomology except that dvol is replaced by the measure w dvol. This paper proves that in the case M = Γ\G/K is a locally symmetric space of nonpositive sectional curvature and arbitrary rank whereupon w(m) = e−dist(m,p0) 2 is the Gaussian relative to some basepoint p0, the w weighted unreduced L2 cohomology is isomorphic to the usual de Rham cohomology. This isomorphism extends to the standard coefficient bundles. Note that weights for the de Rham cohomology of exponential decay have already been constructed; see [Bor83], [Bor90] and [Fra98]. The Gaussian weight behaves differently in terms of coefficient bundles.