Categorical versions of the local Langlands conjecture have been proposed in recent years. The so-called spectral side of the conjecture is the category of coherent sheaves on the moduli stack of local L-parameters (or Galois representations); they are rather concrete objects. The categorical conjecture suggests a rich structure on this category of coherent sheaves, reflecting the Bernstein decomposition of representations of p-adic groups. In a joint work with Alexander Bertoloni Meli, we will construct such a structure directly on the spectral side.
I will first review the categorical conjecture, and outline our project. In the remaining time, I will discuss further details of cuspidal coherent sheaves and their classification.