##### Department of Mathematics,

University of California San Diego

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### Math 209: Number Theory Seminar

## Jessica Fintzen

#### Bonn

## Representations of p-adic groups and Hecke algebras

##### Abstract:

Representations of p-adic groups and Hecke algebras Abstract: An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks, which are indexed by equivalence classes of so called supercuspidal representations of Levi subgroups. In this talk, I will give an overview of what we know about an explicit construction of supercuspidal representations and about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

### January 12, 2024

### 2:00 PM

APM 7321 and Zoom; see https://www.math.ucsd.edu/~nts

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