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### A smooth family of intertwining operators

Raza LAHIANI and Carine MOLITOR-BRAUN
Source: J. Math. Soc. Japan Volume 63, Number 1 (2011), 321-361.

#### 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 πw = indP(w)N χw and π′w = indP′(w)N χw 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.

First Page:
Primary Subjects: 22E30, 22E27
Secondary Subjects: 43A20
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Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1296138354
Digital Object Identifier: doi:10.2969/jmsj/06310321
Zentralblatt MATH identifier: 05871041
Mathematical Reviews number (MathSciNet): MR2752442

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