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

     author = {Hilgert, Joachim},
     title = {A note on Howe's oscillator semigroup},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {39},
     number = {3},
     year = {1989},
     pages = {663-688},
     doi = {10.5802/aif.1182},
     zbl = {0674.47029},
     mrnumber = {91b:22008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_3_663_0}
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/item/AIF_1989__39_3_663_0/

[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 28 #486 | Zbl 0107.09102

[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 88a:22009 | Zbl 0609.22015

[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 83a:81029 | Zbl 0485.22017

[HilHofL] J. Hilgert, K.H. Hofmann and J.D. Lawson, Lie groups, convex cones and semigroups, Oxford University Press, Oxford, 1989. | MR 91k:22020 | Zbl 0701.22001

[How1] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal., 38 (1980), 188-254. | MR 83b:35166 | Zbl 0449.35002

[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 0687.47034

[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 89m:22021 | Zbl 0678.22008

[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 83e:32032 | Zbl 0503.22011

[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 0512.22012

[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 0605.22005

[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 88b:22013 | Zbl 0626.43008