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.

This research supported by the U.M. VIGRE grant.