Norm estimates for unitarizable highest weight modules

Krötz, Bernhard

Annales de l'Institut Fourier, Tome 49 (1999), p. 1241-1264 / Harvested from Numdam

Résumé

Nous considérons des familles de modules de poids dominant unitarisables $({\cal H}_\lambda )_{\lambda \in L}$ sur une demi-droite $L$. Ces modules peuvent être réalisés comme des fonctions holomorphes à valeurs vectorielles sur un domaine borné symétrique ${\cal D}$. Les fonctions polynomiales constituent un sous-ensemble dense de chaque ${\cal H}_\lambda $, $\lambda \in L$. 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 $\lambda \in L$, le module de vecteurs hyperfonctions ${\cal H}_\lambda ^{-\infty }$ peut être réalisé comme l’espace des fonctions holomorphes sur ${\cal D}$.

Abstract

We consider families of unitarizable highest weight modules $({\cal H}_\lambda )_{\lambda \in L}$ on a halfline $L$. All these modules can be realized as vector valued holomorphic functions on a bounded symmetric domain ${\cal D}$, and the polynomial functions form a dense subset of each module ${\cal H}_\lambda $, $\lambda \in L$. 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 $\lambda \in L$ the module of hyperfunction vectors ${\cal H}_\lambda ^{-\infty }$ can be realized as the space of all holomorphic functions on ${\cal D}$.

Détail

Publié le : 1999-01-01

MR 2001i:22013 | Zbl 0930.22013

DOI : https://doi.org/10.5802/aif.1716

Bibliographie

[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. | MR 93m:22016 | Zbl 0785.22014

[ChFa98] H. Chébli, and J. Faraut, Fonctions holomorphes à croissance modérée et vecteurs distributions, submitted. | Zbl 1057.22017

[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 86c:22028 | Zbl 0535.22012

[EJ90] T.J. Enright and A. Joseph, An intrinsic classification of unitary highest weight modules, Math. Ann., 288 (1990), 571-594. | MR 91m:17005 | Zbl 0725.17009

[Fo89] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, New Jersey, 1989. | MR 92k:22017 | Zbl 0682.43001

[Hel78] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Acad. Press, London, 1978. | MR 80k:53081 | Zbl 0451.53038

[HiÓl96] J. Hilgert and G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis, Acad. Press, 1996. | Zbl 0931.53004

[HoTa92] R. Howe and E.C. Tan, Non-Abelian Harmonic Analysis, Springer, New York, Berlin, 1992. | MR 93f:22009 | Zbl 0768.43001

[Jak83] H.P. Jakobsen, Hermitean symmetric spaces and their unitary highest weight modules, J. Funct. Anal., 52 (1983), 385-412. | MR 85a:17004 | Zbl 0517.22014

[Kö69] G. Köthe, Topological Vector Spaces I, Grundlehren der Math. Wissenschaften, 159, Springer, Berlin, Heidelberg, New York, 1969. | Zbl 0179.17001

[KNÓ97] B.K. Krötz, H. Neeb, and G. Ólafsson, Spherical Representations and Mixed Symmetric Spaces, Representation Theory, 1 (1997), 424-461. | MR 99a:22031 | Zbl 0887.22022

[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 96a:22025 | Zbl 0842.22004

[Ne97] K.-H. Neeb, Smooth vectors for highest weight representations, submitted. | Zbl 01546576

[Ne99] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, de Gruyter, to appear. | Zbl 0936.22001

[Sa80] I. Satake, Algebraic Structures of Symmetric Domains, Publications of the Math. Soc. of Japan, 14, Princeton Univ. Press, 1980. | MR 82i:32003 | Zbl 0483.32017

[Tr67] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967. | MR 37 #726 | Zbl 0171.10402