The Plancherel Formula of $L^2(N_0 \setminus G; \psi)$

TL;DR

This paper derives the Plancherel formula for the space of square-integrable functions on a quasi-split p-adic group modulo a unipotent subgroup with a non-degenerate character, decomposing it into generic discrete series representations.

Contribution

It provides the explicit direct integral decomposition and Plancherel formula for $L^2(N_0\setminus G;\psi)$, identifying the discrete spectrum as generic discrete series.

Findings

Discrete spectrum consists of generic discrete series representations.

Explicit Plancherel formula for the space $L^2(N_0\setminus G;\psi)$.

Decomposition into a direct integral of representations.

Abstract

We study the right regular representation on the space $L^2(N_0\setminus G;\psi)$ where $G$ is a quasi-split $p$-adic group and $\psi$ a non-degenerate unitary character of the unipotent subgroup $N_0$ of a minimal parabolic subgroup of $G$. We obtain the direct integral decomposition of this space into its constituent representations. In particular, we deduce that the discrete spectrum of $L^2(N_0\setminus G;\psi)$ consists precisely of $\psi$ generic discrete series representations and derive the Plancherel formula for $L^2(N_0\setminus G;\psi)$.