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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 \(G\).

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E46 Semisimple Lie groups and their representations
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