Geometrization of principal series representations of reductive groups

Details Geometrization of principal series representations of reductive groups Geometrization of principal series representations of reductive groups

2010-11-19

arxiv.org/abs/1011.4529v2

Mathematics

Algebraic Geometry

35 pages. Completely revised in v2; generalizes main results to reductive case and proves what was (Drinfeld's) Conjecture 13;

Scientific paper

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.

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