Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 062, 7 pages      arXiv:0906.2766      https://doi.org/10.3842/SIGMA.2009.062

On the Moore Formula of Compact Nilmanifolds

Hatem Hamrouni
Department of Mathematics, Faculty of Sciences at Sfax, Route Soukra, B.P. 1171, 3000 Sfax, Tunisia

Received December 17, 2008, in final form June 04, 2009; Published online June 15, 2009

Abstract
Let G be a connected and simply connected two-step nilpotent Lie group and Γ a lattice subgroup of G. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation IndΓG(1). Extending then the Abelian case.

Key words: nilpotent Lie group; lattice subgroup; rational structure; unitary representation; Kirillov theory.

pdf (208 kb)   ps (157 kb)   tex (10 kb)

References

  1. Corwin L., Greenleaf F.P., Character formulas and spectra of compact nilmanifolds, J. Funct. Anal. 21 (1976), 123-154.
  2. Corwin L.J., Greenleaf F.P., Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples, Cambridge Studies in Advanced Mathematics, Vol. 18, Cambridge University Press, Cambridge, 1990.
  3. Gelfand I.M., Graev M.I., Piatetski-Shapiro I.I., Representation theory and automorphic functions, W.B. Saunders Co., Philadelphia, Pa.-London - Toronto, Ont. 1969.
  4. Howe R., On Frobenius reciprocity for unipotent algebraic group over Q, Amer. J. Math. 93 (1971), 163-172.
  5. Malcev A.I., On a class of homogeneous spaces, Amer. Math. Soc. Transl. 1951 (1951), no. 39, 33 pages.
  6. Matsushima Y., On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95-110.
  7. Moore C.C., Decomposition of unitary representations defined by discrete subgroups of nilpotent Lie groups, Ann. of Math. (2) 82 (1965), 146-182.
  8. Nielsen O.A., Unitary representations and coadjoint orbits of low dimensional nilpotent Lie groups, Queen's Papers in Pure and Applied Mathematics, Vol. 63, Queen's University, Kingston, ON, 1983.
  9. Onishchik A.L., Vinberg E.B., Lie groups and Lie algebras. II. Discrete subgroups of Lie groups and cohomologies of Lie groups and Lie algebras, Encyclopaedia of Mathematical Sciences, Vol. 21, Springer-Verlag, Berlin, 2000.
  10. Pesce H., Calcul du spectre d'une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339 (1993), 433-461.
  11. Raghunathan M.S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York - Heidelberg, 1972.
  12. Richardson L.F., Decomposition of the L2 space of a general compact nilmanifolds, Amer. J. Math. 93 (1971), 173-190.

Previous article   Next article   Contents of Volume 5 (2009)