Abstract
The results of Kashiwara and Vergne on the decomposition of the tensor products of the ‘Segal-Shale-Weil representation’ are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.
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SegalG. B.: Unitary representations of some infinite dimensional groups,Commun. Math. Phys. 80 (1981), 301–342.
CareyA. L. and RuijsenaarsS. N. M.: On fermion gauge groups, current algebras and Kac-Moody algebras,Acta Appl. Math. 10 (1987), 1–86.
ShaleD.: Linear symmetries of free boson fields,Trans. Amer. Math. Soc. 103 (1962), 149–167.
SchroerB., SeilerR., and SwiecaJ.: Problems of stability for quantum fields in external time-dependent potentials,Phys. Rev. D2 (1970), 2927–2937.
RuijsenaarsS. N. M.: On Bogoliubov transformations II,Ann. of Phys. 116 (1978), 105–134.
MacdonaldI. G.:Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
KashiwaraM. and VergneM.: On the Segal-Shale-Weil representations and harmonic polynomials,Invent. Math. 44 (1978), 1–47.
JakobsenH.: On singular holomorphic representations,Invent. Math. 62 (1980), 67–78.
EnrightT. and ParthasarathyR.: A proof of a conjecture of Kashiwara and Vergne, in J. Carmona and M. Vergne,Non Commutative Harmonic Analysis and Lie Groups, Lecture Notes in Mathematics 880, Springer-Verlag, Berlin, 1981, pp. 74–90.
EnrightT., HoweR., and WallachN.: A classification of unitary highest weight modules, in P. C.Trombi (ed.),Representation Theory of Reductive Groups, Birkhäuser-Verlag, Boston, 1983, pp. 97–143.
JakobsenH.: The last possible place of unitarity for certain highest weight modules,Math. Ann. 256 (1981), 439–447.
JakobsenH.: Hermitian symmetric spaces and their unitary highest weight modules,J. Funct. Anal. 52 (1983), 385–412.
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Schmidt, M.U. Lowest weight representations of some infinite dimensional groups on fock spaces. Acta Appl Math 18, 59–84 (1990). https://doi.org/10.1007/BF00822205
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DOI: https://doi.org/10.1007/BF00822205