  In geometric representation theory, one often wishes to describe
representations realized on spaces of invariant functions as trace functions of
equivariant perverse sheaves. In the case of principal series representations
of a connected split reductive group G over a local field, there is a
description of families of these representations realized on spaces of
functions on G invariant under the translation action of the Iwahori subgroup,
or a suitable smaller compact open subgroup, studied by Howe, Bushnell and
Kutzko, Roche, and others. In this paper, we construct categories of perverse
sheaves whose traces recover the families associated to regular characters of
T(F_q[[t]]), and prove conjectures of Drinfeld on their structure. We also
propose conjectures on the geometrization of families associated to more
general characters.
