Group representations arising from Lorentz conformal geometry

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It is shown that there exist conformally covariant differential operators D2l,k of all even orders 2l, on differential forms of all orders k, in the double cover {A figure is presented}n of the n-dimensional compactified Minkowski space {A figure is presented}n. These act as intertwining differential operators for natural representations of O(2, n), the conformal group of {A figure is presented}n. For even n, the resulting decompositions of differential form representations of O↑(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 2l ≥ |n - 2k|. Physically, these generalize unitary representations of the conformal group associated with the modified wave operator D2,0 = □ + ( (n - 2) 2)2, and the Maxwell operator on vector potentials D2, (n - 2) 2 = δd. All the representation spaces produced, unitary and nonunitary, may be viewed as infinite systems of harmonic oscillators. As a by-product of the spectral resolution of the D2l,k, one gets some striking wave propagative properties for all of the equations D2l,k Φ = 0, including Huygens' principle in the curved spacetime {A figure is presented}n. The operators D2l,k have not been seen before except in the special cases k = 0 or n, and k = (n ± 2) 2, l = 1 (the Maxwell operator). Thus much new information is obtained even in the physical case n = 4. © 1987.




Branson, T. P. (1987). Group representations arising from Lorentz conformal geometry. Journal of Functional Analysis, 74(2), 199–291.

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