Character sheaves and modular representations of finite reductive groups

In this thesis, we study finite reductive groups and their modular representations in non-defining characteristic. In 1990, Geck stated a conjecture on the unitriangularity of decomposition matrices of these groups. Decomposition matrices encode the link between ordinary representations (over a field of characteristic zero) and modular representations (over a field of positive characteristic \( \ell \)). In 2020, Brunat--Dudas--Taylor showed this conjecture for unipotent blocks for a very good prime number \( \ell \), introducing Kawanaka characters. Thanks to the Morita equivalence between unipotent blocks and non-isolated ones, Feng--Späth extended this result to non-isolated blocks in 2021. The aim of this thesis is to study possible generalisations of Brunat--Dudas--Taylor result. Firstly, we extend this result for a bad prime \( \ell \) in the case of simple groups for the unipotents blocks. Inspired by the Brunat--Dudas--Taylor method, we study the decomposition of some Kawanaka characters in terms of ordinary characters in the unipotent blocks. In order to do so, we compute the values of the characteristic functions of characters sheaves on mixed conjugacy classes, based on previous work of Lusztig. Lastly, we show through the examples of \( G_2 \) and \( F_4 \) how the obtained method allows us to study the unitriangularity of isolated blocks for exceptional groups of adjoint types.