Intertwining operators for representations of covering groups of reductive $p$-adic groups

TL;DR

This paper investigates intertwining operators for covering groups of reductive p-adic groups, establishing their adjointness relations and providing explicit formulas for the Harish-Chandra μ-function, which is crucial for understanding representation theory in this context.

Contribution

It introduces a detailed analysis of intertwining operators and derives explicit formulas for the Harish-Chandra μ-function for covering groups of reductive p-adic groups.

Findings

Intertwining operators satisfy specific adjointness relations.

Explicit formula for the Harish-Chandra μ-function in terms of poles and zeros.

Construction of Hermitian forms to locate poles of μ-function.

Abstract

Let $G$ be a covering group of a reductive $p$-adic group. We study intertwining operators between parabolically induced representations of $G$ and prove that they satisfy certain adjointness relations. The Harish-Chandra $\mu$-function is defined as a composition of such intertwining operators for opposite parabolic subgroups of $G$. It can be seen as a complex rational function and we give an explicit formula for it in terms of poles and zeros. The adjointness of the intertwining operators is an important ingredient to prove the formula for the $\mu$-function. To locate the poles of $\mu$, we construct a continuous family of Hermitian forms on a family of parabolically induced representations.