Residues of Intertwining Operators for Classical Groups with an Appendix "$L$-Functions and Poles of Intertwining Operators"

TL;DR

This paper provides an explicit formula for the residues of intertwining operators associated with certain parabolic inductions in symplectic and orthogonal groups over p-adic fields, linking to $L$-functions.

Contribution

It introduces a simple explicit expression for the residues of standard intertwining operators in the context of classical groups with specific Levi subgroup structures.

Findings

Explicit residue formulas for intertwining operators

Connection to $L$-functions and poles

Simplification of previous complex expressions

Abstract

Let $\tilde{G}$ be a symplectic or even orthogonal group over a p-adic field $F$, and $M$ the Levi factor of a maximal parabolic subgroup of $\tilde{G}$. Suppose that $M$ has the shape of three blocks of the same size. Let $\pi$ be a supercuspidal representation of $M$. In this paper we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of $\pi$ from $M$ to $G$.