Norm estimates for unitarizable highest weight modules
Annales de l'Institut Fourier, Volume 49 (1999) no. 4, pp. 1241-1264.

We consider families of unitarizable highest weight modules ( λ ) λL on a halfline L. All these modules can be realized as vector valued holomorphic functions on a bounded symmetric domain 𝒟, and the polynomial functions form a dense subset of each module λ , λL. In this paper we compare the norm of a fixed polynomial in two Hilbert spaces corresponding to two different parameters. As an application we obtain that for all λL the module of hyperfunction vectors λ - can be realized as the space of all holomorphic functions on 𝒟.

Nous considérons des familles de modules de poids dominant unitarisables ( λ ) λL sur une demi-droite L. Ces modules peuvent être réalisés comme des fonctions holomorphes à valeurs vectorielles sur un domaine borné symétrique 𝒟. Les fonctions polynomiales constituent un sous-ensemble dense de chaque λ , λL. Dans ce travail nous comparons les normes d’un polynôme fixé dans deux espaces de Hilbert correspondant à deux paramètres différents. Comme application nous montrons que, pour tout λL, le module de vecteurs hyperfonctions λ - peut être réalisé comme l’espace des fonctions holomorphes sur 𝒟.

@article{AIF_1999__49_4_1241_0,
     author = {Kr\"otz, Bernhard},
     title = {Norm estimates for unitarizable highest weight modules},
     journal = {Annales de l'Institut Fourier},
     pages = {1241--1264},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {4},
     year = {1999},
     doi = {10.5802/aif.1716},
     zbl = {0930.22013},
     mrnumber = {2001i:22013},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1716/}
}
TY  - JOUR
AU  - Krötz, Bernhard
TI  - Norm estimates for unitarizable highest weight modules
JO  - Annales de l'Institut Fourier
PY  - 1999
DA  - 1999///
SP  - 1241
EP  - 1264
VL  - 49
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1716/
UR  - https://zbmath.org/?q=an%3A0930.22013
UR  - https://www.ams.org/mathscinet-getitem?mr=2001i:22013
UR  - https://doi.org/10.5802/aif.1716
DO  - 10.5802/aif.1716
LA  - en
ID  - AIF_1999__49_4_1241_0
ER  - 
%0 Journal Article
%A Krötz, Bernhard
%T Norm estimates for unitarizable highest weight modules
%J Annales de l'Institut Fourier
%D 1999
%P 1241-1264
%V 49
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1716
%R 10.5802/aif.1716
%G en
%F AIF_1999__49_4_1241_0
Krötz, Bernhard. Norm estimates for unitarizable highest weight modules. Annales de l'Institut Fourier, Volume 49 (1999) no. 4, pp. 1241-1264. doi : 10.5802/aif.1716. https://aif.centre-mersenne.org/articles/10.5802/aif.1716/

[BrDe92] J.-L. Brylinski, and P. Delorme, Vecteurs distributions H-Invariants pur les séries principales généralisées d'espaces symétriques réductifs et prolongement méromorphe d'intégrales d'Eisenstein, Invent. Math., 109 (1992), 619-664. | EuDML | MR | Zbl

[ChFa98] H. Chébli, and J. Faraut, Fonctions holomorphes à croissance modérée et vecteurs distributions, submitted. | Zbl

[Cl98] J.-L. Clerc, Distribution vectors for a highest weight representation, submitted.

[EHW83] T.J. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Proc. “Representation theory of reductive groups” (Park City, UT, 1982), 97-149 ; Progr. Math., 40 (1983), 97-143. | MR | Zbl

[EJ90] T.J. Enright and A. Joseph, An intrinsic classification of unitary highest weight modules, Math. Ann., 288 (1990), 571-594. | EuDML | MR | Zbl

[Fo89] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, New Jersey, 1989. | MR | Zbl

[Hel78] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Acad. Press, London, 1978. | MR | Zbl

[HiÓl96] J. Hilgert and G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis, Acad. Press, 1996. | Zbl

[HoTa92] R. Howe and E.C. Tan, Non-Abelian Harmonic Analysis, Springer, New York, Berlin, 1992. | MR | Zbl

[Jak83] H.P. Jakobsen, Hermitean symmetric spaces and their unitary highest weight modules, J. Funct. Anal., 52 (1983), 385-412. | MR | Zbl

[Kö69] G. Köthe, Topological Vector Spaces I, Grundlehren der Math. Wissenschaften, 159, Springer, Berlin, Heidelberg, New York, 1969. | Zbl

[KNÓ97] B.K. Krötz, H. Neeb, and G. Ólafsson, Spherical Representations and Mixed Symmetric Spaces, Representation Theory, 1 (1997), 424-461. | MR | Zbl

[KNÓ98] B.K. Krötz, H. Neeb, and G. Ólafsson, Spherical Functions on Mixed Symmetric Spaces, submitted.

[Ne94a] K.-H. Neeb, Realization of general unitary highest weight representations, Preprint Nr. 1662, TH Darmstadt, 1994.

[Ne94b] K.-H. Neeb, Holomorphic representation theory II, Acta Math., 173:1 (1994), 103-133. | MR | Zbl

[Ne97] K.-H. Neeb, Smooth vectors for highest weight representations, submitted. | Zbl

[Ne99] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, to appear. | Zbl

[Sa80] I. Satake, Algebraic Structures of Symmetric Domains, Publications of the Math. Soc. of Japan, 14, Princeton Univ. Press, 1980. | MR | Zbl

[Tr67] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967. | MR | Zbl

Cited by Sources: