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 $(¥mathfrak{p}(w))_{w¥in ¥mathscr{W}}$ and $(¥mathfrak{p}'(w))_{w¥in ¥mathscr{W}}$ two smooth families of polarizations in $¥mathfrak{n}$. Let π_w = ind_P(w)^Nχ_w and π′_w = ind_P′(w)^Nχ_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.