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)

 
 

 

The heat equation on a compact Lie group


Author: H. D. Fegan
Journal: Trans. Amer. Math. Soc. 246 (1978), 339-357
MSC: Primary 22E30; Secondary 10D20, 58G40
DOI: https://doi.org/10.1090/S0002-9947-1978-0515542-0
MathSciNet review: 515542
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently there has been much work related to Macdonald's $ \eta $-function identities. In the present paper the aim is to give another proof of these identities using analytical methods. This is done by using the heat equation to obtain Kostant's form of the identities. The basic idea of the proof is to look at subgroups of the Lie group which are isomorphic to the group $ SU(2)$. When this has been done the problem has essentially been reduced to that for the group $ SU(2)$, which is a classical result.


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

  • [1] M. Berger, Geometry of the spectrum. I, Proc. Sympos. Pure Math., vol. 27, part 2, Amer. Math. Soc., Providence, R.I., 1975, pp. 129-152. MR 0383459 (52:4340)
  • [2] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin, 1971. MR 0282313 (43:8025)
  • [3] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • [4] F. J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. MR 0522147 (58:25442)
  • [5] H. D. Fegan, The heat equation and modular forms, J. Differential Geometry (to appear). MR 570220 (81k:22006)
  • [6] H. Freudenthal and H. deVries, Linear Lie groups, Academic Press, New York, 1969. MR 0260926 (41:5546)
  • [7] S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565-601. MR 0165032 (29:2323)
  • [8] -, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1-154. MR 0263988 (41:8587)
  • [9] C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Konigsberg, 1829. Reprinted: Gesammelte Werke, Erster Band, Berlin, 1881, pp. 49-239.
  • [10] N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 0143793 (26:1345)
  • [11] B. Kostant, On Macdonald's $ \eta $-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), 179-212. MR 0485661 (58:5484)
  • [12] I. G. Macdonald, Affine root systems and Dedekind's $ \eta $-function, Invent. Math. 15 (1972), 91-143. MR 0357528 (50:9996)
  • [13] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, Cambridge, 1920. MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E30, 10D20, 58G40

Retrieve articles in all journals with MSC: 22E30, 10D20, 58G40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0515542-0
Keywords: Macdonald's identities, 3 dimensional subgroups of a Lie group, Kostant's element ``principal of type $ \rho $"
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society