Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A decomposition for certain real semisimple Lie groups

Author: H. Lee Michelson
Journal: Trans. Amer. Math. Soc. 213 (1975), 177-193
MSC: Primary 22E30; Secondary 43A80
MathSciNet review: 0385002
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition $ G = K{N_0}K$, where $ {N_0}$ is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of $ {N_0}$. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on $ {N_0}$.

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

  • [1] Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 0153782
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [3] Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 0094407
  • [4] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [5] Sigurdur Helgason, A formula for the radial part of the Laplace-Beltrami operator, J. Differential Geometry 6 (1971/72), 411–419. MR 0301460
  • [6] Sigurđur Helgason and Kenneth Johnson, The bounded spherical functions on symmetric spaces, Advances in Math. 3 (1969), 586–593. MR 0249542
  • [7] F. I. Karpelevič, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Trans. Moscow Math. Soc. 1965 (1967), 51–199. Amer. Math. Soc., Providence, R.I., 1967. MR 0231321
  • [8] Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 0364552

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E30, 43A80

Retrieve articles in all journals with MSC: 22E30, 43A80

Additional Information

Article copyright: © Copyright 1975 American Mathematical Society