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

Geometrization of principal series representations of reductive groups