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

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

2011-02-10

arxiv.org/abs/1102.2022v1

Mathematics

Representation Theory

Scientific paper

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 $L^2(N_0\setminus G;\psi)$.

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