On the inner products of some Deligne--Lusztig type representations

In this paper we introduce a family of Deligne–Lusztig type varieties attached to connected reductive groups over quotients of discrete valuation rings, naturally generalising the higher Deligne–Lusztig varieties and some constructions related to the algebraisation problem raised by Lusztig. We establish the inner product formula between the representations associated to these varieties and the higher Deligne–Lusztig representations.