On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Neeb, Karl-H.

doi:10.5802/aif.2660

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
[Vecteurs analytiques pour les représentations unitaires des groups de Lie à dimension infinie]

Neeb, Karl-H.¹

¹ Department Mathematik, FAU Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054-Erlangen, Germany

Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1839-1874.

Résumé
Abstract

Soit $G$ un groupe de Lie–Banach connexe et simplement connexe. Sur l’algèbre enveloppante complexe de son algèbre de Lie $𝔤$ nous définissons la notion de fonctionnelle analytique et montrons que chaque fonctionnelle analytique positive $λ$ est integrable au sens où elle est de la forme $λ (D) = 〈 d π (D) v, v 〉$ pour un vecteur analytique $v$ d’une représentation unitaire de $G$ . Dans la preuve de ce résultat nous obtenons des critères pour l’integrabilité des $*$ -representations des algèbres de Lie en représentations de groupe unitaires.

Pour le coefficient matriciel $π^{v, v} (g) = 〈 π (g) v, v 〉$ d’un vecteur $v$ d’une représentation unitaire d’un groupe de Lie–Fréchet analytique $G$ nous montrons que $v$ est un vecteur analytique si et seulement si $π^{v, v}$ est analytique dans un voisinage de l’identité. En combinant ce résultat à ceux sur les fonctionnelles analytiques positives nous obtenons que chaque fonction analytique de type positive locale sur un group de Lie–Fréchet–BCH simplement connexe s’étend en une fonction analytique globale.

Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $𝔤$ we define the concept of an analytic functional and show that every positive analytic functional $λ$ is integrable in the sense that it is of the form $λ (D) = 〈 d π (D) v, v 〉$ for an analytic vector $v$ of a unitary representation of $G$ . On the way to this result we derive criteria for the integrability of $*$ -representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.

For the matrix coefficient $π^{v, v} (g) = 〈 π (g) v, v 〉$ of a vector $v$ in a unitary representation of an analytic Fréchet–Lie group $G$ we show that $v$ is an analytic vector if and only if $π^{v, v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a simply connected Fréchet–BCH–Lie group $G$ extends to a global analytic function.

MR Zbl

DOI : 10.5802/aif.2660

Classification : 22E65, 22E45
Keywords: Infinite dimensional Lie group, unitary representation, positive definite function, analytic vector, integrability of Lie algebra representations.
Mot clés : groupe de Lie de dimension infinie, représentation unitaire, fonction de type positif, vecteur analytique, integrabilité d’une représentation d’une algèbre de Lie

Affiliations des auteurs :

Neeb, Karl-H. ¹

¹ Department Mathematik, FAU Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054-Erlangen, Germany

@article{AIF_2011__61_5_1839_0,
     author = {Neeb, Karl-H.},
     title = {On {Analytic} {Vectors} for {Unitary} {Representations} of {Infinite} {Dimensional} {Lie} {Groups}},
     journal = {Annales de l'Institut Fourier},
     pages = {1839--1874},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     doi = {10.5802/aif.2660},
     mrnumber = {2961842},
     zbl = {1241.22023},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2660/}
}

TY  - JOUR
AU  - Neeb, Karl-H.
TI  - On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 1839
EP  - 1874
VL  - 61
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2660/
DO  - 10.5802/aif.2660
LA  - en
ID  - AIF_2011__61_5_1839_0
ER  -

%0 Journal Article
%A Neeb, Karl-H.
%T On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups
%J Annales de l'Institut Fourier
%D 2011
%P 1839-1874
%V 61
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2660/
%R 10.5802/aif.2660
%G en
%F AIF_2011__61_5_1839_0

Neeb, Karl-H. On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups. Annales de l'Institut Fourier, Tome 61 (2011) no. 5, pp. 1839-1874. doi : 10.5802/aif.2660. https://aif.centre-mersenne.org/articles/10.5802/aif.2660/

Bibliographie
Cité par

[1] Abouqateb, A.; Neeb, K.-H. Integration of locally exponential Lie algebras of vector fields, Annals Global Analysis Geom., Volume 33:1 (2008), pp. 89-100 | DOI | MR | Zbl

[2] Akhiezer, N. I. The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965

[3] Beltita, D.; Neeb, K.-H. A non-smooth continuous unitary representation of a Banach–Lie group, J. Lie Theory, Volume 18 (2008), pp. 933-936 | MR | Zbl

[4] Berg, C.; Christensen, J.P.R.; Ressel, P. Harmonic analysis on semigroups, Graduate Texts in Math., 100, Springer Verlag, Berlin, Heidelberg, New York, 1984 | MR | Zbl

[5] Bochnak, J.; Siciak, J. Analytic functions in topological vector spaces, Studia Math., Volume 39 (1971), pp. 77-112 | MR | Zbl

[6] Bochnak, J.; Siciak, J. Polynomials and multilinear mappings in topological vector spaces, Studia Math., Volume 39 (1971), pp. 59-76 | MR | Zbl

[7] Borchers, H.-J.; Yngvason, J. Integral representations of Schwinger functionals and the moment problem over nuclear spaces, Comm. math. Phys., Volume 43:3 (1975), pp. 255-271 | DOI | MR | Zbl

[8] Bourbaki, N. Lie Groups and Lie Algebras, Chapter 1–3, Springer Verlag, Berlin, 1989 | MR | Zbl

[9] Bourbaki, N. Espaces vectoriels topologiques. Chap.1 à 5, Springer Verlag, Berlin, 2007

[10] Cartier, P.; Dixmier, J. Vecteurs analytiques dans les représentations de groupes de Lie, Amer. J. Math., Volume 80 (1958), pp. 131-145 | DOI | MR | Zbl

[11] Driver, B.; Gordina, M. Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups (arXiv:math.PR.0809.4979v1)

[12] van Est, W. T.; Korthagen, Th. J. Non enlargible Lie algebras, Proc. Kon. Ned. Acad. v. Wet. Series A, Indag. Math., Volume 26 (1964), pp. 15-31 | MR | Zbl

[13] Faraut, J. Infinite Dimensional Spherical Analysis, COE Lectures Note, 10, Kyushu Univ., 2008 | MR | Zbl

[14] Flato, M.; Simon, J.; Snellman, H.; Sternheimer, D. Simple facts about analytic vectors and integrability, Ann. Sci. École Norm. Sup. (4), Volume 5 (1972), pp. 423-434 | Numdam | MR | Zbl

[15] Gårding, L. Vecteurs analytiques dans les représentations des groupes de Lie, Bull. Soc. Math. France, Volume 88 (1960), pp. 73-93 | Numdam | MR | Zbl

[16] Glöckner, H. Infinite-dimensional Lie groups without completeness restrictions, Geometry and Analysis on Finite and Infinite-dimensional Lie Groups, A. Strasburger, W. Wojtynski, J. Hilgert and K.-H. Neeb (Eds.), Volume 55, Banach Center Publications, 2002, pp. 43-59 | MR | Zbl

[17] Glöckner, H.; Neeb, K.-H. Infinite dimensional Lie groups, Vol. I, Basic Theory and Main Examples (book in preparation)

[18] Goodman, F.; Jørgensen, P. E. T. Lie algebras of unbounded derivations, J. Funct. Anal., Volume 52 (1983), pp. 369-384 | DOI | MR | Zbl

[19] Goodman, R. W. Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc., Volume 143 (1969), pp. 55-76 | DOI | MR | Zbl

[20] Hegerfeldt, G. C. Gårding domains and analytic vectors for quantum fields, J. Math. Phys., Volume 13 (1972), pp. 821-827 | DOI | MR | Zbl

[21] Hegerfeldt, G. C. Extremal decompositions of Wightman functions and of states on nuclear $*$ -algebras by Choquet theory, Comm. Math. Phys., Volume 54:2 (1975), pp. 133-135 | DOI | MR | Zbl

[22] Jørgensen, P. E. T. Operators and Representation Theory, Math. Studies, 147, North-Holland, 1988

[23] Jørgensen, P.E.T. Analytic continuation of local representations of Lie groups, Pac. J. Math., Volume 125:2 (1986), pp. 397-408 | MR | Zbl

[24] Jørgensen, P.E.T. Analytic continuation of local representations of symmetric spaces, J. Funct. Anal., Volume 70 (1987), pp. 304-322 | DOI | MR | Zbl

[25] Jørgensen, P.E.T. Integral representations for locally defined positive definite functions on Lie groups, Int. J. Math., Volume 2:3 (1991), pp. 257-286 | DOI | MR | Zbl

[26] Krein, M. G. Hermitian positive definite kernels on homogeneous spaces I, Amer. Math. Soc. Transl. Ser. 2, Volume 34 (1963), pp. 69-108 | Zbl

[27] Lüscher, M.; Mack, G. Global conformal invariance and quantum field theory, Comm. Math. Phys., Volume 41 (1975), pp. 203-234 | DOI | MR

[28] Magyar, M. Continuous Linear Representations, Math. Studies, 168, North-Holland, 1992 | MR | Zbl

[29] Merigon, S. Integrating representations of Banach–Lie algebras (arXiv:math.RT.1003.0999v1, 4 Mar 2010)

[30] Milnor, J. Remarks on infinite-dimensional Lie groups, in DeWitt, B., Stora, R. (eds), “Relativité, groupes et topologie II” (Les Houches, 1983), North Holland, Amsterdam, 1984; 1007–1057

[31] Moore, R. T. Measurable, continuous and smooth vectors for semigroup and group representations, Memoirs of the Amer. Math. Soc., Volume 19 (1968), pp. 1-80 | MR | Zbl

[32] Müller, C.; Neeb, K.-H.; Seppänen, H. Borel–Weil Theory for Root Graded Banach–Lie groups, Int. Math. Res. Not., Volume 2010:5 (2010), pp. 783-823 | MR | Zbl

[33] Neeb, K.-H. Holomorphy and Convexity in Lie Theory, Exp. in Math. series, 28, de Gruyter Verlag, Berlin, 2000 | MR | Zbl

[34] Neeb, K.-H. Towards a Lie theory of locally convex groups, Jap. J. Math. 3rd ser., Volume 1:2 (2006), pp. 291-468 | MR | Zbl

[35] Neeb, K.-H. On differentiable vectors for representations of infinite dimensional Lie groups, J. Funct. Anal., Volume 259 (2010), pp. 2814-2855 | DOI | MR | Zbl

[36] Nelson, E. Analytic vectors, Annals of Math., Volume 70:3 (1959), pp. 572-615 | DOI | MR | Zbl

[37] Olshanski, G. I.; Vershik, A. M.; Zhelobenko, D. P. Unitary representations of infinite dimensional $(G, K)$ -pairs and the formalism of R. Howe, Representations of Lie Groups and Related Topics, Volume 7, Gordon and Breach Science Publ., 1990 | MR | Zbl

[38] Olshanski, G. I. On semigroups related to infinite dimensional groups, Topics in representation theory, Amer. Math. Soc., , Volume 2, Adv. Sov. mathematics, 1991, pp. 67-101 | MR | Zbl

[39] Powers, R.T. Self-adjoint algebras of unbounded operators, Comm. Math. Phys., Volume 21 (1971), pp. 85-124 | DOI | MR | Zbl

[40] Powers, R.T. Selfadjoint algebras of unbounded operators, II, Trans. Amer. Math. Soc., Volume 187:1 (1974), pp. 261-293 | MR | Zbl

[41] Reed., M. C. A Gårding domain for quantum fields, Comm. Math. Phys., Volume 14 (1969), pp. 336-346 | DOI | MR | Zbl

[42] Reed, S.; Simon, B. Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness, Academic Press, New York, 1975 | Zbl

[43] Rudin, W. Functional Analysis, McGraw Hill, 1973 | MR | Zbl

[44] Samoilenko, Y. S. Spectral Theory of Families of Self-Adjoint Operators, Mathematics and its Applications (Soviet Series), Kluwer Acad. Publ., 1991 | MR

[45] Schmüdgen, K. Positive cones in enveloping algebras, Reports Math. Phys., Volume 14 (1978), pp. 385-404 | DOI | MR | Zbl

[46] Schmüdgen, K. Unbounded Operator Algebras and Representation Theory, Mathematics and its Applications (Soviet Series), 37, Birkhäuser Verlag, Basel, 1990 | MR | Zbl

[47] Shiryaev, A. N. Probability, 2nd Edition, Graduate Texts in Math., 95, Springer, 1996 | MR | Zbl

[48] Simon, J. On the integrability of representations of finite dimensional real Lie algebras, Comm. Math. Phys., Volume 28 (1972), pp. 39-46 | DOI | MR | Zbl

[49] Warner, G. Harmonic analysis on semisimple Lie groups I, Springer Verlag, Berlin, Heidelberg, New York, 1972 | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT