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 IndHGχ. To the character $χ$, we associate the affine subspace X_χ=f+ h^⊥⊂ g^* where f is a linear form associated with $χ$. We describe the irreducible representations occurring in the central decomposition of IndHGχ in terms of the coadjoint orbits in g^* that meet X_χ. We also determine the corresponding measure class on the unitary dual G. Furthermore, we study the multiplicities of the irreducible representations in this decomposition. As an application, we compute the Plancherel formula for IndHGχ when H is a normal algebraic subgroup of G and the multiplicities are finite.