Appears in collection : 2025 - T1 - WS1 - Intertwining operators and geometry

In my talk, I will explain my PhD research project, which is about poles and zeros of the Harish-Chandra $\mu$-function. This function appears in the representation theory of $p$-adic groups, and is defined using intertwining operators between parabolically induced representations. It can be used to describe Bernstein blocks in the category of smooth representations of a reductive $p$-adic group. This work was done by my supervisor Maarten Solleveld, and the goal of my project is to generalize these results to covering groups of reductive $p$-adic groups. To do this, it is necessary to analyze the poles and zeros of the $\mu$-function, which can be seen as a complex rational function. For reductive groups, there is a formula for it given by Silberger, but it is not clear how his proof generalizes to covering groups. Therefore, my supervisor and I have been working on a different proof, which does work for covering groups of reductive $p$-adic groups. The proof uses techniques involving Hermitian and unitary representations, as well as $C^*$-algebras and operator theory. In my talk, I aim to provide the necessary background, before discussing the operator theoretical methods used to locate the poles and zeros of the $\mu$-function.