The scalar product formula for parahoric Deligne--Lusztig induction

Abstract: Parahoric Deligne--Lusztig induction gives rise to positive-depth representations of parahoric subgroups of $p$-adic groups. The most fundamental basic question about parahoric Deligne--Lusztig induction is whether it satisfies the scalar product formula. We resolve this conjecture for all split-generic pairs $(T,\theta)$ -- in particular, for all characters $\theta$ if $T$ is elliptic.