Startseite Mathematik Intertwining operators of irreducible representations for exponential solvable Lie groups
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Intertwining operators of irreducible representations for exponential solvable Lie groups

  • Ali Baklouti EMAIL logo , Hidenori Fujiwara und Jean Ludwig
Veröffentlicht/Copyright: 28. August 2013
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 27 Heft 4

Abstract

Let G be a real solvable exponential Lie group with Lie algebra 𝔤 and let f∈𝔤*. We take two polarizations 𝔭j, j = 1,2, at f which meet the Pukanszky condition. Let Pj:=exp(𝔭j), j = 1,2, be the associated subgroups in G. The linear functional f defines unitary characters χf(expX):=ei〈f,X〉, X∈𝔭j, of Pj. Let πj:=indPjGχf, j = 1,2, be the corresponding induced representations, which are unitary and irreducible. It is well known that τ1 and τ2 are unitarily equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integral I𝔭2,𝔭1. In this paper, we show that this formal integral gives us a concrete intertwining operator provided that it converges absolutely on a G- and 𝒰(𝔤)-invariant dense subspace of the space ℋτ1∞ of C∞ vectors of π1 (which is the case when the simple product P1P2 is closed). Finally, given three Pukanszky polarizations Pi=exp(𝔭i), i = 1,2,3, at f of that kind, we accurately determine the composition formula of I𝔭3,𝔭2∘I𝔭2,𝔭1∘I𝔭1,𝔭3 using Maslov's index.

Keywords: Orbit method; irreducible representations; intertwining operators; Maslov index
MSC: 22E27; 06B15

Funding source: JSPS

Award Identifier / Grant number: 17540180

Funding source: DGRSRT

Award Identifier / Grant number: 00 UR 1501

The authors are deeply grateful to the referee for useful comments in improving the form of the paper.

Received: 2013-1-21
Revised: 2013-6-26
Published Online: 2013-8-28
Published in Print: 2015-7-1

© 2015 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Notes on occupation time fluctuations of binary branching particle systems
  3. On the sum of two integral squares in certain quadratic fields
  4. A Cesàro average of Goldbach numbers
  5. Groups described by element numbers
  6. The second main theorem for entire curves into Hilbert modular surfaces
  7. L∞ norms of holomorphic modular forms in the case of compact quotient
  8. SU(2)-Donaldson invariants of the complex projective plane
  9. Pairs of diagonal quadratic forms and linear correlations among sums of two squares
  10. The subelliptic heat kernel on the three-dimensional solvable Lie groups
  11. Lp boundedness of the commutators of Marcinkiewicz integrals with rough kernels
  12. Circle-valued Morse theory for complex hyperplane arrangements
  13. Atomic Hp spaces and their duals on open subsets of ℝd
  14. Locally contractible coset spaces
  15. L2 and Hp boundedness of strongly singular operators and oscillating operators on Heisenberg groups
  16. Finite time blow-up for semilinear wave equations with variable coefficients
  17. Intertwining operators of irreducible representations for exponential solvable Lie groups
  18. Artin's conjecture and systems of diagonal equations
  19. The homology of simplicial complements and the cohomology of polyhedral products
  20. Multivariate spectral multipliers for the Dunkl transform and the Dunkl harmonic oscillator
  21. Multilinear generalized Radon transforms and point configurations
  22. On ratios of Petersson norms for Yoshida lifts
  23. K-groups for rings of finite Cohen–Macaulay type
  24. Character degree sums of finite groups
  25. Tame Fréchet submanifolds of co-Banach type
  26. Weighted norm inequalities for Schrödinger type operators
https://doi.org/10.1515/forum-2013-6012
Schlagwörter für diesen Artikel
Orbit method; irreducible representations; intertwining operators; Maslov index

Artikel in diesem Heft

  1. Frontmatter
  2. Notes on occupation time fluctuations of binary branching particle systems
  3. On the sum of two integral squares in certain quadratic fields
  4. A Cesàro average of Goldbach numbers
  5. Groups described by element numbers
  6. The second main theorem for entire curves into Hilbert modular surfaces
  7. L∞ norms of holomorphic modular forms in the case of compact quotient
  8. SU(2)-Donaldson invariants of the complex projective plane
  9. Pairs of diagonal quadratic forms and linear correlations among sums of two squares
  10. The subelliptic heat kernel on the three-dimensional solvable Lie groups
  11. Lp boundedness of the commutators of Marcinkiewicz integrals with rough kernels
  12. Circle-valued Morse theory for complex hyperplane arrangements
  13. Atomic Hp spaces and their duals on open subsets of ℝd
  14. Locally contractible coset spaces
  15. L2 and Hp boundedness of strongly singular operators and oscillating operators on Heisenberg groups
  16. Finite time blow-up for semilinear wave equations with variable coefficients
  17. Intertwining operators of irreducible representations for exponential solvable Lie groups
  18. Artin's conjecture and systems of diagonal equations
  19. The homology of simplicial complements and the cohomology of polyhedral products
  20. Multivariate spectral multipliers for the Dunkl transform and the Dunkl harmonic oscillator
  21. Multilinear generalized Radon transforms and point configurations
  22. On ratios of Petersson norms for Yoshida lifts
  23. K-groups for rings of finite Cohen–Macaulay type
  24. Character degree sums of finite groups
  25. Tame Fréchet submanifolds of co-Banach type
  26. Weighted norm inequalities for Schrödinger type operators
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