The heat equation on a compact Lie group

Fegan, H. D.

doi:10.1090/S0002-9947-1978-0515542-0

The heat equation on a compact Lie group
HTML articles powered by AMS MathViewer

by H. D. Fegan PDF

Trans. Amer. Math. Soc. 246 (1978), 339-357 Request permission

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

M. Berger, Geometry of the spectrum. I, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 129–152. MR 0383459
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, DOI 10.1007/BFb0064643
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 [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635–652. MR 522147, DOI 10.1090/S0002-9904-1972-12971-9
H. D. Fegan, The heat equation and modular forms, J. Differential Geometry 13 (1978), no. 4, 589–602 (1979). MR 570220, DOI 10.4310/jdg/1214434710
Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565–601. MR 165032, DOI 10.2307/2373024
Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X

Fundamenta nova theoriae functionum ellipticarum

Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Bertram Kostant, On Macdonald’s $\eta$-function formula, the Laplacian and generalized exponents, Advances in Math. 20 (1976), no. 2, 179–212. MR 485661, DOI 10.1016/0001-8708(76)90186-9
I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931
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, DOI 10.1017/CBO9780511608759