Tokyo Journal of Mathematics

The Multiplicity Function of Mixed Representations on Completely Solvable Lie Groups

Ali BAKLOUTI and Hatem HAMROUNI

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Abstract

We show in this paper that the multiplicities of mixed representations are uniformly infinite or uniformly finite and bounded, in the setting of completely solvable Lie groups extending then the situation of nilpotent Lie groups. Necessary and sufficient conditions for these multiplicities to be finite are provided.

Article information

Source
Tokyo J. Math., Volume 30, Number 1 (2007), 41-55.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1184963646

Digital Object Identifier
doi:10.3836/tjm/1184963646

Mathematical Reviews number (MathSciNet)
MR2328054

Zentralblatt MATH identifier
1151.22011

Subjects
Primary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)

Citation

BAKLOUTI, Ali; HAMROUNI, Hatem. The Multiplicity Function of Mixed Representations on Completely Solvable Lie Groups. Tokyo J. Math. 30 (2007), no. 1, 41--55. doi:10.3836/tjm/1184963646. https://projecteuclid.org/euclid.tjm/1184963646


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References

  • A. Baklouti, A. Ghorbel et H. Hamrouni, Sur les représentations mixtes des groupes de Lie résolubles exponentiels, Publicacions Matemàtiques 46 (2002), 179–199.
  • A. Baklouti and H. Hamrouni, On the down-up representations of exponential solvable Lie groups, Russ. J. Math. Phy. 8 (2001), no. 4, 422–432.
  • R. Benedetti and J. J. Risler, Real algebraic and semi-algebraic sets, Hermann, Paris, 1990.
  • P. Bernat et al., Représentations des groupes de Lie résolubles, Dunod, Paris, 1972.
  • J. Bochnak, M. Coste and M. F. Roy, Géométrie algébrique réelle, Springer-Verlag, 1987.
  • L. Corwin, F. P. Greenleaf and G. Grélaud, Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups, Trans. Amer. Math. Soc. 304 (1987), 537–547.
  • B. N. Currey and R. Penney, The structure of the space of coadjoint orbits of a completely solvable Lie groups, Michigan Math. J. 36 (1989), 309–320.
  • B. N. Currey, The structure of the space of coadjoint orbits on an exponential solvable Lie group, Trans. Am. Math. Soc. 332 (1992), no. 1, 241–269.
  • H. Fujiwara, Représentations monomiales des groupes de Lie résolubles exponentiels, Progress in Math. 82 (1990), 61–84.
  • H. Fujiwara, Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels, Inv. Math. 104 (1991), 647–654.
  • M. E. Herrera, Integration on a semianalytic set, Bull. Soc. Math. France 94 (1966), 141–180.
  • R. Lipsman, The up-down formula for nil-homogeneous spaces, Annali di Matematica Pura ed Applicata (IV),Vol. CLXVI (1994), 291–300.
  • R. Lipsman, Restriction representations of completely solvable Lie groups, Can. J. Math. 42 (1990), 790–824.
  • R. Lipsman, Induced representations of completely solvable Lie groups, Ann. Scuola Norm. Sup. Pisa 17 (1990), 127–164.
  • R. Lipsman, The multiplicity function on exponential and completely solvable homogeneous spaces, Geometriae Dedicata 39 (1991), 155–161.
  • R. Lipsman, Orbital parameters for induced and restricted representations, Trans. Amer. Math. Soc. 313 (1989), 433–473.
  • S. Lojasiewicz, Triangulations of semianalytic sets, Ann. Scuola. Norm. Sup. Pisa 18 (1964), 449–474.
  • J. Risler, Some aspects of complexity in real algebraic geometry, J. Symbolic Computation 5 (1988), 109–119.