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Group representations arising from Lorentz conformal geometry. (English) Zbl 0643.58036
Author’s abstract: “It is shown that there exist conformally covariant differential operators $$D_{2\ell,k}$$ of all even orders $$2\ell$$, on differential forms of all orders k, in the double cover $$M^ n$$ of the n-dimensional compactified Minkowski space $$M^ n$$. These act as intertwining differential operators for natural representations of $$O(2,n)$$, the conformal group of $$M^ n$$. For even n, the resulting decompositions of differential form representations of $$O^{\uparrow}(2,n)$$, the orthochronous conformal group, produce infinite families of unitary representations, the most interesting of which are carried by “positive mass-squared, positive frequency” quotients for $$2\ell \geq | n-2k|$$.”
Reviewer: S.Eloshvili

##### MSC:
 58J90 Applications of PDEs on manifolds 53C80 Applications of global differential geometry to the sciences
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##### References:
 [1] Angelopoulos, E, Sur LES representations unitaires irréductibles de $$SO0 ( p, 2)$$, C.R. acad. sci. Paris, 292, 469-471, (1981) · Zbl 0469.22008 [2] Bargmann, V; Wigner, E.P, Group theoretical discussion of relativistic wave equations, (), 211-223 · Zbl 0030.42306 [3] Bateman, H, The transformation of the electrodynamical equations, (), 223-264 · JFM 41.0942.03 [4] Branson, T, Conformally covariant equations on differential forms, Comm. partial differential equations, 7, 393-431, (1982) · Zbl 0532.53021 [5] Branson, T, Intertwining differential operators for spinor-form representations of the conformal group, Advan. in math., 54, 1-21, (1984) · Zbl 0551.53025 [6] Branson, T, Differential operators canonically associated to a conformal structure, Math. scand., 57, 293-345, (1985) · Zbl 0596.53009 [7] Cunningham, E, The principle of relativity in electrodynamics and an extension thereof, (), 77-98 [8] Dirac, P.A.M, Wave equations in conformal space, Ann. of math., 37, 429-442, (1936) · Zbl 0014.08004 [9] Eisenhart, L.P, Riemannian geometry, (1926), Princeton Univ. Press Princeton, NJ · Zbl 0041.29403 [10] Friedlander, F.G, The wave equation on a curved space-time, (1975), Cambridge Univ. Press Cambridge · Zbl 0316.53021 [11] Gallot, S; Meyer, D, Opérateur de courbure et Laplacian des formes différentielles d’une variété riemannienne, J. math. pures appl., 54, 259-284, (1975) · Zbl 0316.53036 [12] Goodman, R, Analytic and entire vectors for representations of Lie groups, Trans. amer. math. soc., 143, 55-76, (1969) · Zbl 0189.14102 [13] Gross, L, Norm invariance of mass-zero equations under the conformal group, J. math. phys., 5, 687-695, (1964) · Zbl 0127.39306 [14] Helgason, S, Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press New York · Zbl 0451.53038 [15] Helgason, S, Wave equations on homogeneous spaces, (), No. 1077 · Zbl 0146.43601 [16] de Jager, E.M, Theory of distributions, () · Zbl 0148.18302 [17] Jakobsen, H.P; Ørsted, B; Segal, I.E; Speh, B; Vergne, M, Symmetry and causality properties of physical fields, (), 1609-1611 [18] Jakobsen, H.P; Vergne, M, Wave and Dirac operators, and representations of the conformal group, J. funct. anal., 24, 52-106, (1977) · Zbl 0361.22012 [19] Klein, F, Vorlesungen über Höhere geometrie, (1926), Springer-Verlag Berlin · JFM 52.0624.09 [20] Kobayashi, S; Nomizu, K; Kobayashi, S; Nomizu, K, () [21] Kosmann, Y, Sur LES degrés conformes des opérateurs différentiels, C. R. acad. sci. Paris ser. A, 280, 229-232, (1975) · Zbl 0297.53008 [22] Kosmann, Y, Degrés conformes des laplaciens et des opérateurs de Dirac, C. R. acad. sci. Paris ser. A, 280, 283-285, (1975) · Zbl 0296.53012 [23] Kostant, B, Verma modules and the existence of quasi-invariant differential operators, () · Zbl 0372.22009 [24] Lax, P.D; Phillips, R.S, An example of Huygens’ principle, Comm. pure appl. math., 31, 407-416, (1978) [25] Ørsted, B, Wave equations, particles, and chronometric geometry, () [26] Ørsted, B, Conformally invariant differential equations and projective geometry, J. funct. anal., 44, 1-23, (1981) · Zbl 0507.58048 [27] Ørsted, B, The conformal invariance of Huygens’ principle, J. differential geometry, 16, 1-9, (1981) · Zbl 0447.35059 [28] Ørsted, B, Composition series for analytic continuations of holomorphic discrete series representations of SU (n, n), Trans. amer. math. soc., 260, 563-573, (1980) · Zbl 0439.22017 [29] Palais, R.S, A global formulation of the Lie theory of transformation groups, Memoirs amer. math. soc., 22, (1957) · Zbl 0178.26502 [30] Palais, R.S, Foundations of global non-linear analysis, (1968), Benjamin New York · Zbl 0164.11102 [31] Paneitz, S; Segal, I.E, Analysis in space-time bundles. I. general considerations and the scalar bundle, J. funct. anal., 47, 78-142, (1982) · Zbl 0535.58019 [32] Paneitz, S; Segal, I.E, Analysis in space-time bundles. II. the spinor and form bundles, J. funct. anal., 49, 335-414, (1982) · Zbl 0535.58020 [33] Paneitz, S, Analysis in space-time bundles. III. higher spin bundles, J. funct. anal., 54, 18-112, (1983) · Zbl 0535.58021 [34] Paneitz, S, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, (1983), M.I.T, preprint [35] Poulsen, N.S, On C∞-vectors and intertwining bilinear forms for representations of Lie groups, J. funct. anal., 9, 87-120, (1972) · Zbl 0237.22013 [36] Segal, I.E, A class of operator algebras which are determined by groups, Duke math. J., 18, 221-265, (1951) · Zbl 0045.38601 [37] Segal, I.E, Differential operators in the manifold of solutions of a non-linear differential equation, J. math. pures appl., 44, 71-132, (1965) · Zbl 0139.09202 [38] Segal, I.E, Mathematical cosmology and extragalactic astronomy, (1976), Academic Press New York [39] Segal, I.E; Jakobsen, H.P; Ørsted, B; Paneitz, S.M; Speh, B, Covariant chronogeometry and extreme distances: elementary particles, (), 5261-5265 [40] Speh, B, Degenerate series representations of the universal covering group of SU (2, 2), J. funct. anal., 33, 95-118, (1979) · Zbl 0415.22012 [41] Wigner, E.P, On unitary representations of the inhomogeneous Lorentz group, Ann. math., 40, 149-204, (1939) · JFM 65.1129.01 [42] Angelopoulos, E, The unitary irreducible representations of SO0(4, 2), Comm. math. phys., 89, 41-57, (1983) · Zbl 0521.22011 [43] \scT. Branson, Differential form representations of O(p, q), in preparation. [44] Branson, T, Symplectic structure and conserved quantities for some new conformally covariant systems, J. differential equations, 48, 35-59, (1983) · Zbl 0521.58029 [45] Burns, D; Diederich, K; Shnider, S, Distinguished curves in pseudoconvex boundaries, Duke math. J., 44, 407-431, (1977) · Zbl 0382.32011 [46] Enright, T; Howe, R; Wallach, N, A classification of unitary highest weight modules, () · Zbl 0535.22012 [47] \scC. Fefferman and C. R. Graham, Conformal invariants, in “Proceedings of the Symposium: Eli Cartan et les mathématiques d’aujourd’hui,” Asterisque, in press. [48] Jakobsen, H.P, Hermitian symmetric spaces and their unitary highest weight modules, J. funct. anal., 52, 385-412, (1983) · Zbl 0517.22014 [49] Jakobsen, H.P, Conformal covariants, (1985), University of Copenhagen, preprint [50] Knapp, A; Speh, B, Irreducible unitary representations of SU (2, 2), J. funct. anal., 45, 41-73, (1982) · Zbl 0543.22011 [51] Kobayashi, S, Transformation groups in differential geometry, () · Zbl 0246.53031 [52] Molčanov, V.F; Molčanov, V.F, Representations of pseudo-orthogonal groups associated with a cone, Math. USSR sb., Mat. sb., 81, No. 3, 333-347, (1970), Russian original in [53] Penrose, R, Structure of space-time, () · Zbl 1001.83040 [54] Zuckerman, G, Quantum physics and semisimple symmetric spaces, () · Zbl 0554.22005
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