Invariant differential equations on certain semisimple Lie groups
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Abstract:
If G is a semisimple Lie group with one conjugacy class of Cartan subalgebras (e.g. a complex semisimple Lie group), a bi-invariant differential equation on G can be reduced by means of the Radon transform to one on the subgroup MA. In particular, all polynomials of the Casimir operator have a central fundamental solution, and are solvable in ${C^\infty }(G)$; but, for G complex, the “imaginary” Casimir operator is not.References
-
A. Benabdallah, L’opérateur de Casimir de ${\text {SL(2,}}{\textbf {R}})$ (to appear).
- André Cerezo and François Rouvière, Solution élémentaire d’un opérateur différentiel linéaire invariant à gauche sur un groupe de Lie réel compact et sur un espace homogène réductif compact, Ann. Sci. École Norm. Sup. (4) 2 (1969), 561–581 (French). MR 271988, DOI 10.24033/asens.1184 J. Dieudonné, Eléments d’analyse. V, Gauthier-Villars, Paris, 1975.
- Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
- R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 150–165. MR 289724, DOI 10.2307/1970758 V. Guillemin, The Plancherel formula for the complex semisimple Lie groups, Geometric Asymptotics, Math. Surveys, no. 14, Amer. Math. Soc., Providence, R. I., 1977.
- Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- S. Helgason, Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565–601. MR 165032, DOI 10.2307/2373024
- Sigurđur Helgason, An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces, Math. Ann. 165 (1966), 297–308. MR 223497, DOI 10.1007/BF01344014
- Sigurđur Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1–154 (1970). MR 263988, DOI 10.1016/0001-8708(70)90037-X
- Sigurdur Helgason, The surjectivity of invariant differential operators on symmetric spaces. I, Ann. of Math. (2) 98 (1973), 451–479. MR 367562, DOI 10.2307/1970914 L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin, 1963.
- Kenneth D. Johnson, Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups, Nagoya Math. J. 64 (1976), 17–29. MR 480876, DOI 10.1017/S0027763000017529
- Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 364552, DOI 10.24033/asens.1254
- Jeffrey Rauch and David Wigner, Global solvability of the Casimir operator, Ann. of Math. (2) 103 (1976), no. 2, 229–236. MR 425017, DOI 10.2307/1971004
- François Rouvière, Solutions distributions de l’opérateur de Casimir, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 16, Aii, A853–A856. MR 401984
- François Trèves, Linear partial differential equations with constant coefficients: Existence, approximation and regularity of solutions, Mathematics and its Applications, Vol. 6, Gordon and Breach Science Publishers, New York-London-Paris, 1966. MR 0224958
- Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996 G. Warner, Harmonic analysis on semisimple Lie groups. I, Springer-Verlag, Berlin, 1972. D. Želobenko, Harmonic analysis of functions on semisimple Lie groups. II, Math. USSR-Izvestija 3 (1969), 1183-1217.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 243 (1978), 97-114
- MSC: Primary 22E30; Secondary 58G35
- DOI: https://doi.org/10.1090/S0002-9947-1978-0502896-4
- MathSciNet review: 502896