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Norm estimates for unitarizable highest weight modules. (English) Zbl 0930.22013
Let $$\mathbf g$$ be the Lie algebra of the group $$G$$ of automorphisms of an irreducible bounded symmetric domain $${\mathcal D}$$. Then the center $$\mathbf z(\mathbf k)$$ of a maximal compact subalgebra $$\mathbf k$$ of $$\mathbf g$$ is one-dimensional, and for each simple $$\mathbf k$$-module $$F(\lambda)$$ with highest weight $$\lambda$$ one has a natural $$\mathbf g$$-module $$N(\lambda) = {\mathcal U}(\mathbf g_\mathbb C) \otimes_\mathbf q F(\lambda)$$ induced from the extension of the representation of $$\mathbf k$$ on $$F(\lambda)$$ to a maximal parabolic subalgebra $$\mathbf q = \mathbf k_\mathbb C + \mathbf p^+$$ of $$\mathbf g_\mathbb C$$. Fixing the restriction $$F(\lambda_0)$$ to the commutator algebra of $$\mathbf k$$, we obtain a holomorphic family $$F(\lambda_z)$$ of $$\mathbf k$$-modules by tensoring with one-dimensional representations. Under suitable normalizations, the highest weight modules $$N(\lambda_z)$$ are unitary for $$z \in \mathbb R$$ and $$z$$ sufficiently negative. Let $$l(\lambda_0) \subseteq \mathbb R$$ denote the set of all those $$z$$ for which the simple quotient $$L(\lambda_z)$$ of $$N(\lambda_z)$$ is unitary. The main objective of the paper under review is to understand the relation between the scalar products on the modules $$L(\lambda_z)$$ and $$L(\lambda_{z'})$$ if both are unitary. A related (dual) problem is to compare the norms on the corresponding Hilbert spaces $${\mathcal H}_{\lambda_z}$$ of $$F(\lambda_0)$$-valued holomorphic functions on the bounded domain $${\mathcal D}$$. The key observation of this paper is that $$N(\lambda_z)$$ and $$N(\lambda_0)$$ are equivalent as modules of $$[\mathbf k,\mathbf k] + \mathbf p^-$$, where $$\overline{\mathbf q} = \mathbf k_{\mathbb C} + \mathbf p^-$$ is the parabolic subalgebra opposite to $$\mathbf q$$, and that this fact can be used to show that for $$z \leq z'$$ in $$l(\lambda_0)$$ we have $$\langle v,v \rangle_{z'} \leq \langle v,v \rangle_z$$ for $$v \in N(\lambda_z) \cong N(\lambda_0)$$. The opposite relation holds for holomorphic polynomials on $${\mathcal D}$$. As a consequence of these estimates, one obtains a contractive inclusion $${\mathcal H}_{\lambda_{z'}} \to {\mathcal H}_{\lambda_z}$$ of Hilbert spaces and a nice characterization of the hyperfunction vectors for the representation of $$\mathbf g$$ on $${\mathcal H}_\lambda$$ as the closure of the subspace of polynomials in $${\mathcal H}_\lambda$$ in the space of all holomorphic functions $${\mathcal D} \to F(\lambda_0)$$. For the cases where $$N(\lambda_z)$$ is unitary and irreducible, this characterization leads to a simple description of the spherical representations corresponding to compactly causal symmetric spaces of $$G$$ among the unitary highest weight representations of $$G$$.

##### MSC:
 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods 2.2e+47 Semisimple Lie groups and their representations
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##### References:
 [1] 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. · Zbl 0785.22014 [2] H. CHÉBLI, and J. FARAUT, Fonctions holomorphes à croissance modérée et vecteurs distributions, submitted. · Zbl 1057.22017 [3] J.-L. CLERC, Distribution vectors for a highest weight representation, submitted. [4] 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. · Zbl 0535.22012 [5] T.J. ENRIGHT and A. JOSEPH, An intrinsic classification of unitary highest weight modules, Math. Ann., 288 (1990), 571-594. · Zbl 0725.17009 [6] G.B. FOLLAND, Harmonic analysis in phase space, Princeton University Press, Princeton, New Jersey, 1989. · Zbl 0682.43001 [7] S. HELGASON, Differential geometry, Lie groups, and symmetric spaces, Acad. Press, London, 1978. · Zbl 0451.53038 [8] J. HILGERT and G. ÓLAFSSON, Causal symmetric spaces, geometry and harmonic analysis, Acad. Press, 1996. · Zbl 0931.53004 [9] R. HOWE and E.C. TAN, Non-abelian harmonic analysis, Springer, New York, Berlin, 1992. · Zbl 0768.43001 [10] H.P. JAKOBSEN, Hermitean symmetric spaces and their unitary highest weight modules, J. Funct. Anal., 52 (1983), 385-412. · Zbl 0517.22014 [11] G. KÖTHE, Topological vector spaces I, Grundlehren der Math. Wissenschaften, 159, Springer, Berlin, Heidelberg, New York, 1969. · Zbl 0179.17001 [12] B.K. KRÖTZ, H. NEEB, and G. ÓLAFSSON, Spherical representations and mixed symmetric spaces, Representation Theory, 1 (1997), 424-461. · Zbl 0887.22022 [13] B.K. KRÖTZ, H. NEEB, and G. ÓLAFSSON, Spherical functions on mixed symmetric spaces, submitted. [14] K.-H. NEEB, Realization of general unitary highest weight representations, Preprint Nr. 1662, TH Darmstadt, 1994. [15] K.-H. NEEB, Holomorphic representation theory II, Acta Math., 173:1 (1994), 103-133. · Zbl 0842.22004 [16] K.-H. NEEB, Smooth vectors for highest weight representations, submitted. · Zbl 1029.17007 [17] K.-H. NEEB, Holomorphy and convexity in Lie theory, Expositions in Mathematics, de Gruyter, to appear. · Zbl 0936.22001 [18] I. SATAKE, Algebraic structures of symmetric domains, Publications of the Math. Soc. of Japan, 14, Princeton Univ. Press, 1980. · Zbl 0483.32017 [19] F. TREVES, Topological vector spaces, Distributions and Kernels, Academic Press, New York, London, 1967. · Zbl 0171.10402
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