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

Author: Roberto J. Miatello
Journal: Trans. Amer. Math. Soc. 260 (1980), 1-33
MSC: Primary 58G11; Secondary 10D40, 22E40, 43A85, 53C35, 58G25
Erratum: Trans. Amer. Math. Soc. 264 (1981), 587.
MathSciNet review: 570777
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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.

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  • [ABP] M. Atiyah, R. Bott and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330. MR 0650828 (58:31287)
  • [BGM] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971. MR 0282313 (43:8025)
  • [CW] R. Cahn and J. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comment. Math. Helv. 51 (1976), 1-21. MR 0397801 (53:1657)
  • [CGW] R. Cahn, P. Gilkey and J. Wolf, Heat equation, proportionality principle, and volume of fundamental domains, Math. Phys. Appl. Math., vol. 3, Reidel, Dordrecht, 1976, pp. 43-54. MR 0436222 (55:9170)
  • [C] O. Campoli, The complex Fourier transform on rank one semisimple Lie groups, Thesis, Rutgers University, 1976.
  • [G] D. De George, Length spectrum for compact locally symmetric spaces of strictly negative curvature, Ann. École Norm. Sup. 10 (1977), 133-152. MR 461863 (81i:58047)
  • [GWa] D. De George and N. Wallach, Limit formulas for multiplicits in $ {L^2}(\Gamma \backslash G)$. II. The tempered spectrum, Ann. of Math. (to appear).
  • [Ga] R. Gangolli, Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151-192. MR 0239000 (39:360)
  • [GW] R. Gangolli and G. Warner, On Selberg's trace formula, J. Math. Soc. Japan 27 (1975), 328-343. MR 0399354 (53:3199)
  • [GGP] I. M. Gel'fand, M. Graev and I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, Pa., 1969. MR 0233772 (38:2093)
  • [HC1] Harish-Chandra, Discrete series for semisimple Lie groups. II, Acta Math. 116 (1966), 1-111. MR 0219666 (36:2745)
  • [HC2] -, Harmonie analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), 117-201. MR 0439994 (55:12875)
  • [HS] H. Hecht and W. Schmid, A proof of Blattner's conjecture, Invent. Math. 31 (1975), 129-154. MR 0396855 (53:715)
  • [He] S. Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1-154. MR 0263988 (41:8587)
  • [J] K. Johnson, Differential equations and an analog of the Paley Wiener theorem for linear semisimple Lie groups, Nagoya Math. J. 64 (1976), 17-29. MR 0480876 (58:1025)
  • [KS] A. Knapp and E. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489-578. MR 0460543 (57:536)
  • [Ko] B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627-642. MR 0245725 (39:7031)
  • [L] R. Langlands, On the classification of irreducible representations of algebraic groups, Institute for Advanced Study, Princeton, N.J., 1975 (preprint).
  • [MS] H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), 43-69. MR 0217739 (36:828)
  • [M] R. Miatello, On the Plancherel measure for linear Lie groups of rank one, Manuscripta Math. 29 (1979), 249-276. MR 545044 (80h:22021)
  • [MP] S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256. MR 0031145 (11:108b)
  • [P] R. Palais et al., Seminar on the Atiyah-Singer index theorem, Ann. of Math. Studies, vol. 59, Princeton Univ. Press, Princeton, N.J., 1965. MR 0198494 (33:6649)
  • [R] M. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, Berlin and New York, 1972. MR 0507234 (58:22394a)
  • [Wa1] N. R. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)
  • [Wa2] -, On the Selberg trace formula in the case of compact quotient, Bull. Amer. Math. Soc. 82 (1976), 171-195. MR 0404533 (53:8333)
  • [Wa3] -, On an asymptotic formula of Gelfand and Gangolli for the spectrum of $ \Gamma \backslash G$, J. Differential Geometry 11 (1976), 91-101. MR 0417340 (54:5396)
  • [Wa4] -, On Harish-Chandra's generalized c-functions, Amer. J. Math. 97 (1975), 386-403. MR 0399357 (53:3202)
  • [War] G. Warner, Harmonic analysis on semisimple Lie groups. I, Springer-Verlag, Berlin and New York, 1972.

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