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 split-generic pairs (T, ) -- in particular, for all characters if T is elliptic.