  A module over an affine Kac--Moody algebra g^ is called spherical if the
action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of
the corresponding group G[[t]]. Consider the category of spherical g^-modules
of critical level. In this paper we prove that this category is equivalent to
the category of quasi-coherent sheaves on the ind-scheme of opers on the
punctured disc which are unramified as local systems. This result is a
categorical version of the well-known description of spherical vectors in
representations of groups over local non-archimedian fields. It may be viewed
as a special case of the local geometric Langlands correspondence proposed in
arXiv:math/0508382.
