Journal of the Mathematical Society of Japan

A smooth family of intertwining operators

Raza LAHIANI and Carine MOLITOR-BRAUN

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1296138354

Digital Object Identifier
doi:10.2969/jmsj/06310321

Mathematical Reviews number (MathSciNet)
MR2752442

Zentralblatt MATH identifier
1231.22012

Subjects
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.

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

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


Export citation

References

  • A. Baklouti, H. Fujiwara and J. Ludwig, Intertwining operators of irreducible representations for exponential solvable Lie groups, preprint.
  • A. Baklouti and J. Ludwig, Entrelacement des restrictions des représentations unitaires des groupes de Lie nilpotents, Annales de l'Institut Fourier, 51 (2001), 395–429.
  • L. Corwin and P. Greenleaf, Representations of nilpotent Lie groups and their applications, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, 1990.
  • A. Didier, H. Fujiwara and J. Ludwig, Opérateurs d'entrelacement pour les groupes de Lie exponentiels, Amer. J. Math., 118 (1996), 839–878.
  • J. Dieudonné, Eléments d'analyse 2, Gauthier-Villars, Paris, 1969.
  • H. Fujiwara, Certains opérateurs d'entrelacement pour des groupes de Lie résolubles exponentiels et leurs applications, Mem. Fac. Sci., Kyushu Univ., 36 (1982), 13–72.
  • R. Howe, On a connection between nilpotent groups and oscillatory integrals associated to singularities, Pacific J. Math., 73 (1977), 329–363.
  • J. Howie, Complex Analysis, Springer undergraduate mathematics series, 2003.
  • R. Lahiani, Analyse harmonique sur certains groupes de Lie à croissance polynomiale, PhD thesis, University of Luxembourg/University of Metz, March 2010.
  • H. Leptin and J. Ludwig, Unitary Representation Theory of Exponential Lie Groups, de Gruyter Expositions in Mathematics, 18, Walter de Gruyter & Co., Berlin, 1994.
  • G. Lion, Intégrales d'entrelacement sur des groupes de Lie nilpotents et indices de Maslov, Lectures Notes in Math., 587, Springer-Verlag, 1977, pp.,160–176.
  • J. Ludwig, C. Molitor-Braun and L. Scuto, On Fourier's inversion theorem in the context of nilpotent Lie groups, Acta Sci. Math., 73 (2007), 547–591.
  • J. Ludwig and D. Müller, Sub-Laplacians of holomorphic $L^{p}$-type on rank one $AN$-groups and related solvable groups, J. Funct. Anal., 170 (2000), 366–427.
  • J. Ludwig and H. Zahir, On the nilpotent *-Fourier transform, Lett. Math. Phys., 30 (1994), 23–34.
  • V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Mathematics, 102, Springer-Verlag, 1984.