  The $L^2$-cohomology of a locally symmetric variety is known to have the
topological interpretation as the intersection homology of its Baily-Borel
Satake compactification. In this article, we observe that even without the
Hermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$
finite and sufficiently large, is isomorphic to the ordinary cohomology of its
reductive Borel-Serre compactification. We use this to generalize a theorem of
Mumford concerning homogeneous vector bundles, their invariant Chern forms and
the canonical extensions of the bundles; here, though, we are referring to
canonical extensions to the reductive Borel-Serre compactification of any
arithmetic quotient. To achieve that, we give a systematic discussion of vector
bundles and Chern classes on stratified
