Spherical functions on Cartan motion groups

Rader, Cary

doi:10.1090/S0002-9947-1988-0965746-4

Spherical functions on Cartan motion groups
HTML articles powered by AMS MathViewer

by Cary Rader PDF

Trans. Amer. Math. Soc. 310 (1988), 1-45 Request permission

Abstract:

This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general $K$-type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of $K$ in the universal enveloping algebra of the corresponding reductive group.

References

Jean-Louis Clerc, Estimations à l’infini des fonctions des Bessel généralisées, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 8, 429–430 (French, with English summary). MR 611408
Jean-Louis Clerc, Estimations à l’infini des fonctions des Bessel généralisées, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 8, 429–430 (French, with English summary). MR 611408
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math. 49 (1983), no. 3, 309–398. MR 707179
J. M. G. Fell, The dual spaces of Banach algebras, Trans. Amer. Math. Soc. 114 (1965), 227–250. MR 172131, DOI 10.1090/S0002-9947-1965-0172131-9
J. M. G. Fell, Non-unitary dual spaces of groups, Acta Math. 114 (1965), 267–310. MR 186754, DOI 10.1007/BF02391824
Roger Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc. 73 (1952), 496–556. MR 52444, DOI 10.1090/S0002-9947-1952-0052444-2
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Sigurdur Helgason, A duality for symmetric spaces with applications to group representations. III. Tangent space analysis, Adv. in Math. 36 (1980), no. 3, 297–323. MR 577307, DOI 10.1016/0001-8708(80)90019-5
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
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
Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 342641
B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI 10.2307/2373470
J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1–44. MR 346093, DOI 10.1090/S0002-9947-1973-0346093-X
Anders Melin and Johannes Sjöstrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) Lecture Notes in Math., Vol. 459, Springer, Berlin, 1975, pp. 120–223. MR 0431289
Cary Rader, Spherical functions on a semi-simple Lie group, University of Chicago, Department of Mathematics, Chicago, Ill., 1976. Lecture notes in representation theory. MR 0498992
Shôichirô Sakai, On the representations of semi-simple Lie groups, Proc. Japan Acad. 30 (1954), 14–18. MR 63374

Linear partial differential equations with constant coefficients

Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158

Harmonic analysis on semi-simple Lie groups