Representation theory of quantized enveloping algebras and Hecke algebras

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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.