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

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.

New in second version: section 4 is improved and divided into subsections. Sections 4.2 and 4.3 were added to study the limit of the mu-function at z=0 and z=infinity, this was necessary to complete the proof of the formula for the mu-function