The scalar product formula for parahoric Deligne--Lusztig induction

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 Howe-factorizable split-generic pairs $(T,\theta )$ – in particular, for all characters $\theta$ when $T$ is elliptic and $p$ is not a torsion prime for the root system of the $p$-adic group.