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

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