  The geometric Langlands program is distinguished in assigning spectral
decompositions to all representations, not only the irreducible ones. However,
it is not even clear what is meant by a spectral decomposition when one works
with non-abelian reductive groups and with ramification. The present work is
meant to clarify the situation, showing that the two obvious candidates for
such a notion coincide.
  More broadly, we study the moduli space of (possibly irregular) de Rham local
systems from the perspective of homological algebra. We show that, in spite of
its infinite-dimensional nature, this moduli space shares many of the nice
features of an Artin stack. Less broadly, these results support the belief in
the existence of a version of geometric Langlands allowing arbitrary
ramification.
  Along the way, we give some apparently new, if unsurprising, results about
the algebraic geometry of the moduli space of connections, using
Babbitt-Varadarajan's reduction theory for differential equations.
