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

TL;DR

This paper introduces a new family of Deligne--Lusztig type varieties for reductive groups over certain rings, establishing inner product formulas with existing higher Deligne--Lusztig representations, advancing understanding in algebraic representation theory.

Contribution

It generalizes higher Deligne--Lusztig varieties and provides inner product formulas linking new and existing representations, addressing algebraisation issues.

Findings

Established inner product formulas for the new representations

Generalized higher Deligne--Lusztig varieties to new algebraic contexts

Connected the new varieties with classical representation theory

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.