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References
A. Borel, Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963), 111–122.
E. P. van den Ban, Asymptotic expansions and integral formulas for eigenfunctions on a semisimple Lie group, Thesis, Utrecht, 1982.
W. Casselman and D. Miličic, Asymptotic behaviour of matrix coefficients of admissible representations. Duke Math. J. 49 (1982), 869–930.
J. Chazarain, Formule de Poisson pour les variétés Riemanniennes. Inv. Math. 24 (1974), 65–82.
_____, Spectre des opérateurs elliptiques et flots hamiltoniens. Seminaire Bourbaki Exposé No. 460, 111–123. Springer Lecture Notes in Mathematics No. 514, Springer-Verlag, Berlin, 1976.
D. DeGeorge, Length spectrum for compact locally symmetric spaces of strictly negative curvature. Ann. Scient. École Norm. Sup. (4) 10 (1977), 133–152.
J. J. Duistermaat, Oscillatory integrals, Lagrange immersions, and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974), 207–281.
J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature. Inv. Math. 52 (1979), 29–93.
_____, Functions, flows, and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. To appear in Compositio Math.
J. J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math. 29 (1975), 37–79.
R. A. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups. Ann. Math. 93 (1971), 150–165.
_____, Beta functions of Selberg's type for compact space forms of symmetric spaces of rank one. Ill. J. Math. 21 (1977), 1–41.
I. M. Gel'fand, Automorphic functions and the theory of representations. Proc. Int. Congress of Math. Stockholm 1962, 74–85.
V. Guillemin and S. Sternberg, Geometric asymptotics. Mathematical surveys No. 14, American Mathematical Society, Providence, R.I., 1977.
Harish-Chandra, Spherical functions on a semisimple Lie group, I. Amer. J. Math. 80 (1958), 241–310.
_____, Spherical functions on a semisimple Lie group, II. Amer. J. Math. 80 (1958), 553–613.
_____, Some results on differential equations and their applications. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1763–1764.
_____, Some results on differential equations. Preprint, 1960.
S. Helgason, An analogue of the Paley-Wiener theorem for Fourier transform on certain symmetric spaces. Math. Annalen, 165 (1966), 297–308.
D. Hejhal, The Selberg trace formula for PSL(2,ℝ), vol. I. Springer Lecture Notes in Math. No. 548, Springer-Verlag, Berlin, 1976.
S. Helgason and K. Johnson, The bounded spherical functions on symmetric spaces. Advances in Math. 3 (1969), 586–593.
L. Hörmander, The spectral function of an elliptic operator. Acta Math. 121 (1968), 193–218.
P. D. Lax and R. S. Phillips, Scattering theory for automorphic functions. Ann. Mathematics. Studies. No. 87, Princeton University Press, Princeton, N.J., 1976.
J. Millson, Closed geodesics and the η-invariant. Ann. Math. 108 (1978), 1–39.
B. Randol, A Dirichlet series of eigenvalue type with applications to asymptotic estimates. Bull. Lond. Math. Soc. 13 (1981), 309–315.
P. Sarnak, Class numbers of indefinite binary quadratic forms. J. Number Theory 15 (1983), 229–247.
_____, The arithmetic and geometry of some hyperbolic three manifolds. Preprint.
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Ind. Math. Soc. 20 (1956), 47–87.
I. M. Singer, Eigenvalues of the Laplacian and invariants of manifolds. Proc. Int. Cong. Math. Vancouver, 1974, 187–200.
R. J. Stanton and P. A. Tomas, Expansions for spherical functions on noncompact symmetric spaces. Acta Math. 140 (1978), 251–276.
P. C. Trombi and V. S. Varadarajan, Spherical transforms on semisimple Lie groups. Ann. Math. 94 (1971), 246–303.
V. S. Varadarajan, Lie groups, Lie algebras, and their representations. Prentice-Hall, Englewood Cliffs, N.J., 1974.
_____, Harmonic analysis on real reductive groups. Springer Lecture Notes in Math. No. 576, Springer-Verlag, Berlin, 1977.
_____, Oscillatory integrals and their applications to harmonic analysis on real semisimple Lie groups. To appear in the Proceedings of the conference on reductive groups held at Utah, Spring 1982.
N. Wallach, On the Selberg trace formula in the case of compact quotient. Bull. A.M.S. 82 (1976), 171–195.
H. Weyl, Ramifications, old and new, of the eigenvalue problem. Bull. A.M.S. 56 (1950), 115–139.
S. T. Yau, Seminar on differential geometry. Ann. Mathematics studies No. 102, Princeton University Press, Princeton, N.J., 1982.
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Varadarajan, V.S. (1984). Asymptotic properties of eigenvalues and eigenfunctions of invariant differential operators on symmetric and locally symmetric spaces. In: Herb, R., Johnson, R., Lipsman, R., Rosenberg, J. (eds) Lie Group Representations III. Lecture Notes in Mathematics, vol 1077. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072347
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