## Journal of the Mathematical Society of Japan

### 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 $\pi_w = \mathsf{ind}_{P(w)}^N \chi_w$ and $\pi'_w = \mathsf{ind}_{P'(w)}^N \chi_w$ be the corresponding induced representations, which are unitary and irreducible. It is well known that $\pi_w$ and $\pi'_w$ 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 63, Number 1 (2011), 321-361.

Dates
First available in Project Euclid: 27 January 2011

https://projecteuclid.org/euclid.jmsj/1296138354

Digital Object Identifier
doi:10.2969/jmsj/06310321

Mathematical Reviews number (MathSciNet)
MR2752442

Zentralblatt MATH identifier
1231.22012

#### Citation

LAHIANI, Raza; MOLITOR-BRAUN, Carine. A smooth family of intertwining operators. J. Math. Soc. Japan 63 (2011), no. 1, 321--361. doi:10.2969/jmsj/06310321. https://projecteuclid.org/euclid.jmsj/1296138354

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