Representations of P-Adic Groups and Hecke Algebras

Explore the intricate world of p-adic group representations and Hecke algebras in this comprehensive lecture. Delve into the category of smooth, complex representations of p-adic groups and its significance in constructing explicit or categorical local Langlands correspondences. Examine the decomposition of representation categories into Bernstein blocks, indexed by equivalence classes of supercuspidal representations of Levi subgroups. Gain insights into the explicit construction of supercuspidal representations and the structure of Bernstein blocks. Learn about ongoing research that aims to establish equivalences between general Bernstein blocks and depth-zero Bernstein blocks through Hecke algebra isomorphisms. This one-hour talk provides a thorough overview of current knowledge and ongoing developments in the field of p-adic group representations.