Norm estimates for unitarizable highest weight modules

Krötz, Bernhard

Annales de l'Institut Fourier, Tome 49 (1999) no. 4, pp. 1241-1264.

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

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

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