Cohomological representations of parahoric subgroups

Abstract

We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric.