On Hecke algebras for $p$-adic group

In this talk, I will present two results concerning the Hecke algebras for $p$-adic groups. First, I will discuss a result that, under mild tameness assumptions, every Bernstein block of a $p$-adic group $G$ is equivalent to a depth-zero Bernstein block of a twisted Levi subgroup of $G$. Moreover, these blocks are also equivalent to the category of modules over an extension of an affine Hecke algebra by a twisted group algebra. This result is the main result of my joint work with Jeffrey D. Adler, Jessica Fintzen, and Manish Mishra.