Kyoto Journal of Mathematics

The notion of cusp forms for a class of reductive symmetric spaces of split rank 1

Erik P. van den Ban, Job J. Kuit, and Henrik Schlichtkrull

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study a notion of cusp forms for the symmetric spaces G/H with G=SL(n,R) and H=S(GL(n1,R)×GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series representations of G/H coincides with the space of cusp forms.

Article information

Kyoto J. Math., Volume 59, Number 2 (2019), 471-513.

Received: 26 January 2016
Accepted: 3 April 2017
First available in Project Euclid: 9 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 22E46: Semisimple Lie groups and their representations 43A80: Analysis on other specific Lie groups [See also 22Exx]

cusp form cuspidal integral reductive symmetric space discrete series


van den Ban, Erik P.; Kuit, Job J.; Schlichtkrull, Henrik. The notion of cusp forms for a class of reductive symmetric spaces of split rank $1$. Kyoto J. Math. 59 (2019), no. 2, 471--513. doi:10.1215/21562261-2019-0015.

Export citation


  • [1] N. B. Andersen, M. Flensted-Jensen, and H. Schlichtkrull, Cuspidal discrete series for semisimple symmetric spaces, J. Funct. Anal. 263 (2012), no. 8, 2384–2408.
  • [2] D. Bălibanu and E. P. van den Ban, Convexity theorems for semisimple symmetric spaces, Forum Math. 28 (2016), no. 6, 1167–1204.
  • [3] E. P. van den Ban, The principal series for a reductive symmetric space, II: Eisenstein integrals, J. Funct. Anal. 109 (1992), no. 2, 331–441.
  • [4] E. P. van den Ban and J. J. Kuit, Cusp forms for reductive symmetric spaces of split rank one, Represent. Theory 21 (2017), 467–533.
  • [5] Y. Benoist, Analyse harmonique sur les espaces symétriques nilpotents, J. Funct. Anal. 59 (1984), no. 2, 211–253.
  • [6] G. van Dijk and M. Poel, The Plancherel formula for the pseudo-Riemannian space $\mathrm{SL}(n,\mathbf{R})/{\mathrm{GL}}(n-1,\mathbf{R})$, Compos. Math. 58 (1986), no. 3, 371–397.
  • [7] G. van Dijk and M. Poel, The irreducible unitary $\mathrm{GL}(n-1,\mathbf{R})$-spherical representations of $\mathrm{SL}(n,\mathbf{R})$, Compos. Math. 73 (1990), no. 1, 1–30.
  • [8] S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math. 12, Academic Press, New York, 1962.
  • [9] A. G. Helminck and S. P. Wang, On rationality properties of involutions of reductive groups, Adv. Math. 99 (1993), no. 1, 26–96.
  • [10] A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser Boston, Boston, 2002.
  • [11] M. T. Kosters and G. van Dijk, Spherical distributions on the pseudo-Riemannian space $\mathrm{SL}(n,\mathbf{R})/{\mathrm{GL}}(n-1,\mathbf{R})$, J. Funct. Anal. 68 (1986), no. 2, 168–213.
  • [12] W. A. Kosters, Eigenspaces of the Laplace-Beltrami-operator on $\mathrm{SL}(n,\mathbf{R})/{\mathrm{S}}(\mathrm{GL}(1)\times\mathrm{GL}(n-1))$, I, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 1, 99–123. II, no. 2, 125–145.
  • [13] O. Loos, Symmetric Spaces, I: General Theory, W. A. Benjamin, New York, 1969.
  • [14] T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), no. 2, 331–357.
  • [15] G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200–221.
  • [16] H. Ochiai, Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one, Indag. Math. (N.S.) 16 (2005), nos. 3–4, 631–638.
  • [17] W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), no. 1, 157–180.