The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature

Miatello, Roberto J.

doi:10.1090/S0002-9947-1980-0570777-5

The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature
HTML articles powered by AMS MathViewer

by Roberto J. Miatello PDF

Trans. Amer. Math. Soc. 260 (1980), 1-33 Request permission

Erratum: Trans. Amer. Math. Soc. 264 (1981), 587.

Abstract:

We use harmonic analysis on semisimple Lie groups to determine the Minakshisundaram-Pleijel asymptotic expansion for the trace of the heat kernel on natural vector bundles over compact, locally symmetric spaces of strictly negative curvature.

References

M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR 650828, DOI 10.1007/BF01425417
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
Robert S. Cahn and Joseph A. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comment. Math. Helv. 51 (1976), no. 1, 1–21. MR 397801, DOI 10.1007/BF02568140
Robert S. Cahn, Peter B. Gilkey, and Joseph A. Wolf, Heat equation, proportionality principle, and volume of fundamental domains, Differential geometry and relativity, Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, 1976, pp. 43–54. MR 0436222

The complex Fourier transform on rank one semisimple Lie groups

David L. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly negative curvature, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 133–152. MR 461863, DOI 10.24033/asens.1323

Limit formulas for multiplicits in

The tempered spectrum

R. Gangolli, Asymptotic behavior of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151–192. MR 239000, DOI 10.1007/BF02391912
Ramesh Gangolli and Garth Warner, On Selberg’s trace formula, J. Math. Soc. Japan 27 (1975), 328–343. MR 399354, DOI 10.2969/jmsj/02720328
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 0233772
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR 439994, DOI 10.2307/1971058
Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112
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
Kenneth D. Johnson, Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups, Nagoya Math. J. 64 (1976), 17–29. MR 480876, DOI 10.1017/S0027763000017529
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4

On the classification of irreducible representations of algebraic groups

H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
Roberto J. Miatello, On the Plancherel measure for linear Lie groups of rank one, Manuscripta Math. 29 (1979), no. 2-4, 249–276. MR 545044, DOI 10.1007/BF01303630
S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242–256. MR 31145, DOI 10.4153/cjm-1949-021-5
Richard S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. MR 0198494, DOI 10.1515/9781400882045
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
Nolan R. Wallach, On the Selberg trace formula in the case of compact quotient, Bull. Amer. Math. Soc. 82 (1976), no. 2, 171–195. MR 404533, DOI 10.1090/S0002-9904-1976-13979-1
Nolan R. Wallach, An asymptotic formula of Gelfand and Gangolli for the spectrum of $G\backslash G$, J. Differential Geometry 11 (1976), no. 1, 91–101. MR 417340
Nolan R. Wallach, On Harish-Chandra’s generalized $C$-functions, Amer. J. Math. 97 (1975), 386–403. MR 399357, DOI 10.2307/2373718

Harmonic analysis on semisimple Lie groups