Tang U-Liang

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

We study the right regular representation on the space $L^2(N_0\setminus G;ψ)$ where $G$ is a quasi-split $p$-adic group and $ψ$ 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 $L^2(N_0\setminus G;ψ)$ consists precisely of $ψ$ generic discrete series representations and derive the Plancherel formula for $L^2(N_0\setminus G;ψ)$.