The Selberg trace formula. V. Questions of trace class

Osborne, M. Scott; Warner, Garth

doi:10.1090/S0002-9947-1984-0756044-9

The Selberg trace formula. V. Questions of trace class
HTML articles powered by AMS MathViewer

by M. Scott Osborne and Garth Warner PDF

Trans. Amer. Math. Soc. 286 (1984), 351-376 Request permission

Abstract:

The purpose of this paper is to develop criteria which will ensure that the $K$-finite elements of $C_c^\infty (G)$ are represented on $L_{{\text {dis}}}^2(G/\Gamma )$ by trace class operators.

References

James Arthur, The trace formula for reductive groups, Conference on automorphic theory (Dijon, 1981) Publ. Math. Univ. Paris VII, vol. 15, Univ. Paris VII, Paris, 1983, pp. 1–41. MR 723181, DOI 10.1007/978-1-4684-6730-7_{1}
Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
Siu Yuen Cheng, Peter Li, and Shing Tung Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), no. 5, 1021–1063. MR 630777, DOI 10.2307/2374257
Harold Donnelly, On the point spectrum for finite volume symmetric spaces of negative curvature, Comm. Partial Differential Equations 6 (1981), no. 9, 963–992. MR 627655, DOI 10.1080/03605308108820201
Harold Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Differential Geometry 17 (1982), no. 2, 239–253. MR 664496
Harold Donnelly and Peter Li, Lower bounds for the eigenvalues of negatively curved manifolds, Math. Z. 172 (1980), no. 1, 29–40. MR 576293, DOI 10.1007/BF01182776
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181, DOI 10.1007/BFb0079929
Werner Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197–288. MR 725778, DOI 10.1002/mana.19831110109
M. Scott Osborne and Garth Warner, The Selberg trace formula. I. $\Gamma$-rank one lattices, J. Reine Angew. Math. 324 (1981), 1–113. MR 614517, DOI 10.1515/crll.1981.324.1
M. Scott Osborne and Garth Warner, The Selberg trace formula. II. Partition, reduction, truncation, Pacific J. Math. 106 (1983), no. 2, 307–496. MR 699915, DOI 10.2140/pjm.1983.106.307
M. Scott Osborne and Garth Warner, The Selberg trace formula. II. Partition, reduction, truncation, Pacific J. Math. 106 (1983), no. 2, 307–496. MR 699915, DOI 10.2140/pjm.1983.106.307

The Selberg trace formula

Atle Selberg, Discontinuous groups and harmonic analysis, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 177–189. MR 0176097
Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983), no. 1, 48–79. MR 705991, DOI 10.1016/0022-1236(83)90090-3
N. R. Wallach, On the constant term of a square integrable automorphic form, Operator algebras and group representations, Vol. II (Neptun, 1980) Monogr. Stud. Math., vol. 18, Pitman, Boston, MA, 1984, pp. 227–237. MR 733320
Garth Warner, Selberg’s trace formula for nonuniform lattices: the $R$-rank one case, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 1–142. MR 535763