Representation theory of quantized enveloping algebras and Hecke algebras

群是代数结构，与物体、公式、结构中的对称性有关。它们出现在许多上下文中。为了更好地理解它们，人们需要具体地认识群体，这是通过表征理论来实现的。这些表示可以被分解成更简单的表示，即不可约的表示。它们是表征理论的基石。量子化包络代数和Hecke代数也是一种很好的泛化群的代数抽象对象。它们都出现在李论领域。更准确地说，当人们对如何将一个表示分割成一些群的不可约表示的问题感兴趣时，赫克代数自然出现了，更准确地说，是在有限的p进域上定义的可约群。这些赫克代数的表示需要分析这种分裂。其中，细胞表征起着重要作用。当人们想要研究同一群的不可约模表示时，也会出现量子化包络代数（或量子群）。在这两种情况下，这些代数的表示都可以借助D. Kazhdan和G.Lusztig对Hecke代数以及M.Kashiwara和G.Lusztig对量子化包膜代数发现的正则基来研究。这些基具有重要的性质，为表示提供了很好的实现。然而，关于他们的许多事情仍然是神秘的。在本研究计划中，我们将研究这些规范基以及量子化包络代数和Hecke代数的细胞表示。

Groups are algebraic structures that are related to symmetry in objects, in formulas, in structures. They appear in many context. To better understand them, one is led to realize groups concretely and  this is done through representation theory.  The representations can be broken up in simpler representations that are said to be irreducible. They are the building blocks of representation theory.    Quantized enveloping algebras and Hecke algebras are also algebraic abstract objects that generalize groups in a nice manner. They both appear in the field of Lie theory. More precisely,  Hecke algebras appear naturally when one is interested in the problem of  how to split a representation into irreducible representations for some groups, more precisely reductive groups defined over finite ou p-adic fields. The representations of these Hecke algebras are needed to analyse this splitting.  Among these, the cells representations play an essential role.  Quantized enveloping algebras (or quantum groups) also appear when one want to study  irreducible modular representation for the same groups. In both cases, the representations of these algebras can be studied with the help of canonical  bases  discovered by D. Kazhdan and G. Lusztig for Hecke algebras,  and by M.Kashiwara and G.Lusztig for quantized enveloping algebras. These bases have important properties and give good realization for the representations. However, much about them  is  still mysterious. In this research program, we will study these canonical bases and the cells representations for quantized enveloping algebras and Hecke algebras.