Cohomological representations of parahoric subgroups

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