Harmonic Analysis on $p$-adic Unipotent Groups

Let $G$ be a $p$-adic unipotent group and let $H$ be an algebraic subgroup of $G$. Given a unitary character $χ$ of $H$, we consider the monomial representation $\operatorname{Ind}_{H}^{G}χ$. To the character $χ$, we associate the affine subspace $$ X_χ=f+\mathfrak h^\perp\subset\mathfrak g^*$$ where $f$ is a linear form associated with $χ$. We describe the irreducible representations occurring in the central decomposition of $\operatorname{Ind}_{H}^{G}χ$ in terms of the coadjoint orbits in $\mathfrak g^*$ that meet $X_χ$. We also determine the corresponding measure class on the unitary dual $\widehat G$. Furthermore, we study the multiplicities of the irreducible representations in this decomposition. As an application, we compute the Plancherel formula for $\operatorname{Ind}_{H}^{G}χ$ when $H$ is a normal algebraic subgroup of $G$ and the multiplicities are finite.