A note on Howe's oscillator semigroup
Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 663-688.

Analytic extensions of the metaplectic representation by integral operators of Gaussian type have been calculated in the L 2 ( n ) and the Bargmann-Fock realisations by Howe [How2] and Brunet-Kramer [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]], respectively. In this paper we show that the resulting semigroups of operators are isomorphic and calculate the intertwining operator.

Brunet, Kramer et Howe ont établi l’existence des continuations analytiques pour la représentation métaplectique par des semigroupes d’opérateurs intégraux dans L 2 ( n ) (voir [Howe, Proc. Symp. Pure Math., 48 (1988)] et dans l’espace de Fock (voir [Brunet-Kramer, Reports on Math. Phys., 17 (1980), 205-215]). Dans cet article on démontre que les deux semigroupes sont isomorphes et on détermine l’opérateur d’entrelacement.

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     title = {A note on {Howe's} oscillator semigroup},
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Hilgert, Joachim. A note on Howe's oscillator semigroup. Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 663-688. doi : 10.5802/aif.1182. http://www.numdam.org/articles/10.5802/aif.1182/

[Ba1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure. Appl. Math., 14 (1961), 187-214. | MR | Zbl

[Ba2] V. Bargmann, Group representations on Hilbert spaces of analytic functions, in Analytic Methods in Mathematical Physics, Gilbert and Newton, Eds. Gordon and Breach, New York, 1968.

[Br] M. Brunet, The metaplectic semigroup and related topics, Reports on Math. Phys., 22 (1985), 149-170. | MR | Zbl

[BrK] M. Brunet and P. Kramer, Complex extension of the representation of the symplectic group associated with the canonical commutation relations, Reports on Math. Phys., 17 (1980), 205-215. | MR | Zbl

[HilHofL] J. Hilgert, K.H. Hofmann and J.D. Lawson, Lie groups, convex cones and semigroups, Oxford University Press, Oxford, 1989. | MR | Zbl

[How1] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal., 38 (1980), 188-254. | MR | Zbl

[How2] R. Howe, The oscillator semigroup, in the mathematical heritage of Hermann Weyl, Proc. Symp. Pure Math., 48, R.O. Wells, Ed. AMS Providence, 1988. | Zbl

[K] P. Kramer, Composite particles and symplectic (semi)-groups, in Group Theoretical Methods in Physics, P. Kramer and A. Rieckers Ed., LNP, 79, Springer, Berlin, 1978.

[KMS] P. Kramer, M. Moshinsky and T.H. Seligman, Complex extensions of canonical transformations and quantum mechanics, in Group theory and its applications III, E. Loeble Ed., Acad. Press, New York, 1975.

[LM] M. Lüscher and G. Mack, Global conformal invariance in quantum field theory, Comm. Math. Phys., 41 (1975), 203-234.

[OlaØ] G. 'Olafsson and B. Ørsted, The holomorphic discrete series for affine symmetric spaces I, J. Funct. Anal., 81 (1988), 126-159. | MR | Zbl

[Ol'1] G.I. Ol'Shanskii, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funct. Anal. and Appl., 15 (1981), 275-285. | MR | Zbl

[Ol'2] G.I. Ol'Shanskii, Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Sov. Math. Dokl., 26 (1982), 97-101. | Zbl

[Ol'3] G.I. Ol'Shanskii, Unitary representations of the infinite symmetric group : a semigroup approach in Representations of Lie groups and Lie algebras, Akad. Kiado, Budapest, 1985. | Zbl

[R] L.J.M. Rothkrantz, Transformatiehalfgroepen van nietcompacte hermitesche symmetrische Ruimten, Dissertation, Univ. of Amsterdam, 1980.

[S] R.J. Stanton, Analytic extension of the holomorphic discrete series, Amer. J. of Math., 108 (1986), 1411-1424. | MR | Zbl

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