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
MathSciNet review: 515542
Full-text PDF Free Access

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, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 129–152. MR 0383459
  • [2] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
  • [3] 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
  • [4] Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652. MR 0522147, 10.1090/S0002-9904-1972-12971-9
  • [5] H. D. Fegan, The heat equation and modular forms, J. Differential Geom. 13 (1978), no. 4, 589–602 (1979). MR 570220
  • [6] Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
  • [7] S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565–601. MR 0165032
  • [8] Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 0263988
  • [9] C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Konigsberg, 1829. Reprinted: Gesammelte Werke, Erster Band, Berlin, 1881, pp. 49-239.
  • [10] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [11] Bertram Kostant, On Macdonald’s 𝜂-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179–212. MR 0485661
  • [12] I. G. Macdonald, Affine root systems and Dedekind’s 𝜂-function, Invent. Math. 15 (1972), 91–143. MR 0357528
  • [13] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469

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: http://dx.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