Title: On the inner products of some Deligne--Lusztig type representations,
Abstract: 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.