Generalized pseudo-coefficients of discrete series of $p$-adic groups

TL;DR

This paper constructs generalized pseudo-coefficients for discrete series representations of Levi subgroups in p-adic groups, extending classical concepts and applying them to the Plancherel formula for harmonic analysis.

Contribution

It introduces a new class of functions that generalize pseudo-coefficients for discrete series, facilitating harmonic analysis on p-adic groups.

Findings

Existence of locally constant, compactly supported functions generalizing pseudo-coefficients.

Application of these functions to the Plancherel formula.

Extension of pseudo-coefficient theory to Levi subgroups.

Abstract

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $\sigma$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $\sigma.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.