Introduction to the Schwartz Space of T\G

Casselman, W.

Canadian journal of mathematics, Tome 41 (1989) no. 2, pp. 285-320

Résumé

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G.This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

Détail

DOI : 10.4153/CJM-1989-015-6

Bibliographie

[1] 1. Arthur, J., A trace formula for reductive groups I: terms associated to classes in G(Q), Duke Math. Jour. 45(1978), 911–952. Google Scholar

[2] 2. Borel, A., Reduction theory for arithmetic groups pp. 20–25 in Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math. 9 (American Mathematical Society, Providence, 1966). Google Scholar

[3] 3. Introduction to automorphic forms, pp. 199–210 in Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math. 9 (American Mathematical Society, Providence, 1966). Google Scholar

[4] 4. Borel, A., Introduction aux groupes arithmétiques (Hermann, Paris, 1969). Google Scholar

[5] 5. Borel, A., Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geom. 6 (1972), 543–560. Google Scholar

[6] 6. Borel, A., Représentations de groupes localement compact, Lecture Notes in Mathematics 276 (Springer-Verlag, New York, 1972). Google Scholar

[7] 7. Borel, A., Stable real cohomology of arithmetic groups, Ann. scient. Ecole Norm. Sup. 7 (1974), 235–272. Google Scholar

[8] 8. Borel, A., A decomposition theorem for functions of uniform moderate growth on T\CL preprint, Institute for Advanced Study, (1983). Google Scholar

[9] 9. Borel, A. and Jacquet, H., Automorphicforms andautomorphic representations, pp. 189–202 in Automorphic forms, representations, and L-functions, Proc. Symp. Pure Math. 33 (American Mathematical Society, Providence, 1979). Google Scholar

[10] 10. Borel, A. and Serre, J-P., Corners and arithmetic groups, Comm. Math. Helv. 48 (1973), 436–491. Google Scholar

[11] 11. Borel, A. and Tits, J., Groupes réductifs, Publ. Math. Inst. Hautes Etudes Sci. 27 (1965), 55–120. Google Scholar

[12] 12. Casselman, W.A., Canonical extensions of Harish-Chandra modules to representations of C, preprint (1987). Google Scholar

[13] 13. Dixmier, J. and Malliavin, P., Factorisations de fonctions et de vecteurs indéfiniment différentiahles, Bull. Sci. Math. 102 (1978), 307–330. Google Scholar

[14] 14. Duistermaat, J.J., Fourier integral operators, Lecture notes published by the Courant Institute of Mathematical Sciences, New York (1974). Google Scholar

[15] 15. Godement, R., The spectral decomposition of cusp forms in Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math. 9 (American Mathematical Society, Providence, 1966). Google Scholar

[16] 16. Gelfand, I.M. and Piatetski-Shapiro, I.I., Automorphic functions and representation theory, Trudy Moskov. Mat. Obsc. 12 (1963), 389–412. Google Scholar

[17] 17. Harish-Chandra, , Automorphic forms on semi-simple Lie groups, Lecture Notes in Mathematics 62 (Springer-Verlag, New York, 1968). Google Scholar

[18] 18. Harish-Chandra, , Harmonic analysis on semi-simple Lie groups, Bull. Amer. Math. Soc. 76 (1970), 529–551. Google Scholar

[19] 19. Landau, E., Einige Ungleichungen für zweimal differentierbare Funktionen, Proc. London Math. Soc. 13 (1914), 43–49. Google Scholar

[20] 20. Langlands, R.P., letter to A. Borel, dated October 25, 1972. Google Scholar

[21] 21. Langlands, R.P., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics 544 (Springer-Verlag, New York, 1976). Google Scholar

[22] 22. Raghunathan, , Discrete subgroups of Lie groups, Ergebnisse der Mathematik 68 (Springer-Verlag, New York, 1972). Google Scholar

[23] 23. Satake, I., On compactifications of the quotient spaces for arithmetically defined discontinuous subgroups, Ann. Math. 72 (1960), 555–580. Google Scholar

[24] 24. Treves, F., Topological vector spaces, distributions, and kernels, (Academic Press, New York, 1967). Google Scholar

[25] 25. Wallach, N., Asymptotic expansions of generalized matrix entries of representations oj real reductive groups, in Lie group representations I (Proceedings, University of Maryland 1982-1983), Lecture Notes in Mathematics 1024 (Springer-Verlag, New York, 1983). Google Scholar

[26] 26. Warner, G., Harmonic analysis on semi-simple groups, vols. I and II, Grundlehren fiir Math. Wiss. 188–189, (Springer-Verlag, New York, 1972). Google Scholar

[27] 27. Zucker, S., Is-cohomology of warped products and arithmetic groups, Inv. Math. 70(1982), 169–218. Google Scholar