| Résume | In this talk, I will explain two results concerning the representations of $p$-adic groups. First, I will present a result showing that, under mild tameness assumptions, every indecomposable direct factor (called a Bernstein block) of the category of representations of a $p$-adic group is equivalent to the category of modules over a certain modification of an affine Hecke algebra. As a corollary, every Bernstein block is equivalent to a special type of Bernstein block, called a depth-zero block, which is closely related to representations of finite groups of Lie type and is therefore much more accessible than general Bernstein blocks. This is the main
result of my joint work with Jeffrey D. Adler, Jessica Fintzen, and Manish Mishra. Second, I will explain how the computation of the $q$-parameters of the above affine Hecke algebra reduces to the unipotent case, where explicit computations were obtained by George Lusztig. This result can be regarded as a part of ``reduction to unipotent'' result. |
