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