本项目旨在研究离散赋值环上李型群的几何表示论。令G为任意离散赋值环上的约化群概型的有理点子群（即李型群），我们将关注G的光滑表示。一方面，Lusztig从几何角度出发，利用有限域上代数簇的平展上同调构造了这类群的一类非常有趣的表示（称为高阶Deligne--Lusztig表示），另一方面，Gerardin从代数角度出发，利用轨迹方法和Clifford理论亦构造了这类群的许多表示；这两类构造方法在一定条件下有相同的参数集，我们将沿着这些参数对比分析两边的表示，特别的，我们期待能在许多情形下证明它们的同构性。在此之后，我们计划利用这些结果来研究高阶Deligne--Lusztig表示的基本性质，如本原性和半单性，以及这些性质在有限域上李代数的应用。最后，我们希望研究基于这些构造的特征标层理论。

This project aims to study the geometric representation theory of Lie type groups over discrete valuation rings. Let G be the rational point subgroup of a reductive group scheme over a discrete valuation ring (that is, a group of Lie type); we will focus on the smooth representations of G. On the one hand, Lusztig has constructed a family of very interesting representations of these groups from the geomtric aspect, by using etale cohomology of varieties over finite fields, on the other hand, Gerardin has also constructed many representations of these groups from the algebraic aspect, by using orbit method and Clifford theory; under certain conditions, the two constructions share the same set of parametres, and we will compare and analyse the two families of representations along these parametres, in particular, we expect to prove their isomorphicity in many situations. After then, we plan to use these results to study the fundamental properties of higher Deligne--Lusztig representations, like primitivity and semisimplicity, as well as their applications to the Lie algebras over finite fields. In the final part, we hope to study the theory of character sheaves based on these constructions.