A smooth family of intertwining operators

概要

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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 πw = indP(w)Nχw and π′w = indP′(w)Nχw be the corresponding induced representations, which are unitary and irreducible. It is well known that πw and π′w are unitary equivalent. In this paper, we prove the existence of a smooth family of intertwining operator (Tw)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.