Fourier algebras of parabolic subgroups


  • Søren Knudby



We study the following question: given a locally compact group when does its Fourier algebra coincide with the subalgebra of the Fourier-Stieltjes algebra consisting of functions vanishing at infinity? We provide sufficient conditions for this to be the case.

As an application, we show that when $P$ is the minimal parabolic subgroup in one of the classical simple Lie groups of real rank one or the exceptional such group, then the Fourier algebra of $P$ coincides with the subalgebra of the Fourier-Stieltjes algebra of $P$ consisting of functions vanishing at infinity. In particular, the regular representation of $P$ decomposes as a direct sum of irreducible representations although $P$ is not compact.


Baggett, L., Unimodularity and atomic Plancherel measure, Math. Ann. 266 (1984), no. 4, 513–518.

Baggett, L. and Taylor, K., Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), no. 3, 593–600.

Baggett, L. and Taylor, K., A sufficient condition for the complete reducibility of the regular representation, J. Funct. Anal. 34 (1979), no. 2, 250–265.

Cowling, M. and Haagerup, U., Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549.

Dixmier, J., Sur les représentations unitaires des groupes de Lie algébriques, Ann. Inst. Fourier, Grenoble 7 (1957), 315–328.

Dixmier, J., Sur les représentations unitaires des groupes de Lie nilpotents. V, Bull. Soc. Math. France 87 (1959), 65–79.

Dixmier, J., $C^*$-algebras, North-Holland Mathematical Library, vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

Eymard, P., L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236.

Figà-Talamanca, A., Positive definite functions which vanish at infinity, Pacific J. Math. 69 (1977), no. 2, 355–363.

Folland, G. B., A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

Ghandehari, M., Amenability properties of Rajchman algebras, Indiana Univ. Math. J. 61 (2012), no. 3, 1369–1392.

Haagerup, U. and Knudby, S., The weak Haagerup property II: Examples, Int. Math. Res. Not. IMRN (2015), no. 16, 6941–6967.

Hewitt, E. and Zuckerman, H. S., Singular measures with absolutely continuous convolution squares, Proc. Cambridge Philos. Soc. 62 (1966), 399–420.

Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, corrected reprint of the 1986 original ed., Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997.

Kaniuth, E., Lau, A. T., and Ülger, A., The Rajchman algebra $B_0(G)$ of a locally compact group $G$, Bull. Sci. Math. 140 (2016), no. 3, 273–302.

Kaniuth, E. and Taylor, K. F., Induced representations of locally compact groups, Cambridge Tracts in Mathematics, vol. 197, Cambridge University Press, Cambridge, 2013.

Keene, F. W., Lipsman, R. L., and Wolf, J. A., The Plancherel formula for parabolic subgroups, Israel J. Math. 28 (1977), no. 1-2, 68–90.

Khalil, I., Sur l'analyse harmonique du groupe affine de la droite, Studia Math. 51 (1974), 139–167.

Kleppner, A. and Lipsman, R. L., The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132.

Knapp, A. W., Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002.

Mackey, G. W., Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311.

Mauceri, G., Square integrable representations and the Fourier algebra of a unimodular group, Pacific J. Math. 73 (1977), no. 1, 143–154.

Menchoff, D., Sur l'unicité du développement trigonométrique, C. R. Acad. Sci., Paris 163 (1916), 433–436.

Nachbin, L., The Haar integral, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965.

Pier, J.-P., Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984.

Takahashi, R., Quelques résultats sur l'analyse harmonique dans l'espace symétrique non compact de rang $1$ du type exceptionnel, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II, Lecture Notes in Math., vol. 739, Springer, Berlin, 1979, pp. 511--567.

Taylor, K. F., Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983), no. 2, 183–190.

Taylor, K. F., Groups with atomic regular representation, Representations, wavelets, and frames, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2008, pp. 33--45.

Walter, M. E., On a theorem of Figà-Talamanca, Proc. Amer. Math. Soc. 60 (1976), 72–74 (1977).

Wolf, J. A., Representations of certain semidirect product groups, J. Functional Analysis 19 (1975), no. 4, 339–372.




How to Cite

Knudby, S. (2017). Fourier algebras of parabolic subgroups. MATHEMATICA SCANDINAVICA, 120(2), 272–290.