Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A quick proof of Harish-Chandra's Plancherel theorem for spherical functions on a semisimple Lie group


Author: Jonathan Rosenberg
Journal: Proc. Amer. Math. Soc. 63 (1977), 143-149
MSC: Primary 22E30; Secondary 43A90
DOI: https://doi.org/10.1090/S0002-9939-1977-0507231-8
MathSciNet review: 0507231
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Some lemmas of S. Helgason and R. Gangolli, originally conceived for proving an analogue of the Paley-Wiener theorem for symmetric spaces, are used to give a quick proof of Harish-Chandra's inversion formula and Plancherel theorem for bi-invariant functions on a semisimple Lie group. The method is elementary in that it does not require introduction of Harish-Chandra's ``Schwartz space."


References [Enhancements On Off] (What's this?)

  • [1] R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 150-165. MR 44 #6912. MR 0289724 (44:6912)
  • [2] -, Spherical functions on semisimple Lie groups, Symmetric Spaces, (W. Boothby and G. Weiss, Editors), Dekker, New York, 1972. MR 0420157 (54:8172)
  • [3] S. G. Gindikin and F. I. Karpelevič, Plancherel measure for Riemann symmetric spaces of nonpositive curvature, Dokl. Akad. Nauk SSSR 145 (1962), 252-255 = Soviet Math. Dokl. 3 (1962), 962-965. MR 27 #240. MR 0150239 (27:240)
  • [4] Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241-310. MR 20 #925. MR 0094407 (20:925)
  • [5] -, Spherical functions on a semisimple Lie group. II, Amer. J. Math. 80 (1958), 553-613. MR 21 #92. MR 0101279 (21:92)
  • [6] S. Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297-308. MR 36 #6545. MR 0223497 (36:6545)
  • [7] -, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1-154. MR 41 #8587. MR 0263988 (41:8587)
  • [8] -, Analysis on Lie groups and homogeneous spaces, CBMS Regional Conf. Ser. in Math., no. 14, Amer. Math. Soc., Providence, R. I., 1972. MR 47 #5179. MR 0316632 (47:5179)
  • [9] G. Warner, Harmonic analysis on semisimple Lie groups. II, Springer-Verlag, Berlin and New York, 1972.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22E30, 43A90

Retrieve articles in all journals with MSC: 22E30, 43A90


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0507231-8
Keywords: Semisimple Lie group, spherical function, inversion formula, Plancherel theorem, c-function
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society