Journal of the Mathematical Society of Japan

A smooth family of intertwining operators


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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 $\pi_w = \mathsf{ind}_{P(w)}^N \chi_w$ 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

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

First available in Project Euclid: 27 January 2011

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Zentralblatt MATH identifier

Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc.

smooth intertwining operators smooth Malcev and Jordan-Holder bases generalized Kirillov result generalized Schwartz spaces


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.

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