A smooth family of intertwining operators

Abstract

Let N be a connected, simply connected nilpotent Lie group with Lie algebra $\mathfrak{n}$ and let $\mathscr{W}$ be a submanifold of $\mathfrak{n}$^* such that the dimension of all polarizations associated to elements of $\mathscr{W}$ is fixed. We choose ($\mathfrak{p}$(w))_{w ∈ $\mathscr{W}$} and ($\mathfrak{p}$′(w))_{w ∈ $\mathscr{W}$} two smooth families of polarizations in $\mathfrak{n}$. Let π_w = ind_P(w)^N χ_w and π′_w = ind_P′(w)^N χ_w be the corresponding induced representations, which are unitary and irreducible. It is well known that π_w and π′_w are unitary equivalent. In this paper, we prove the existence of a smooth family of intertwining operator (T_w)_w for theses representations, where w runs through an appropriate non-empty relatively open subset of $\mathscr{W}$. The intertwining operators are given by an explicit formula.