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F-method for symmetry breaking operators. (English) Zbl 1311.22016
The author first gives geometric criteria for finite multiplicity and uniformly bounded multiplicity of the local and non-local symmetric breaking operators, and next, by extending the so-called F-method known for local operators to non-local ones, he obtains the functional identities and the explicit residue formulas of regular symmetric breaking operators on rank one orthogonal groups.
Let \(G\) be a real reductive linear Lie group and \(P\) a parabolic subgroup of \(G\). For a finite dimensional representation \(\lambda\) of \(P\) on \(V\), let \({\mathcal V}=G\times_P V\) be the associated homogeneous vector bundle over \(G/P\). Then \(G\) acts continuously on the space \(C^\infty(G/P,{\mathcal V})\) of smooth sections. Similarly, we define \(C^\infty(G'/P',{\mathcal W})\) for a reductive subgroup \(G'\) of \(G\), a parabolic subgroup \(P'\) of \(G'\), and a finite dimensional representation of \(P'\) on \(W\). We assume \(P'\subset P\cap G\). Then the space of (non-local) symmetric breaking operators is given by \[ \text{Hom}_{G'}(C^\infty(G/P,{\mathcal V}), C^\infty(G'/P',{\mathcal W})). \] As a subspace of \(G'\)-intertwining differential operators, the local symmetric breaking operators is given by \[ \text{Diff}_{G'}(C^\infty(G/P,{\mathcal V}), C^\infty(G'/P',{\mathcal W})). \] For these spaces, geometric equivalent conditions that \(\dim <\infty\) (finite multiplicity) and \(\sup_{V}\sup_{W}\dim <\infty\) (uniformly bounded multiplicity) are obtained (Theorems 2.3 and 2.7). Let \({\mathfrak g}={\mathfrak n}^- + {\mathfrak l} + {\mathfrak n}^+\) be the Gelfand-Naimark decomposition of the complexification \(\mathfrak g\) of the Lie algebra \({\mathfrak g}_{\mathbb R}\) of \(G\). By extending F-method to non-local operators, the Fourier transform from \({\mathcal S}'({\mathfrak n}_{\mathbb R}^-)\) to \({\mathcal S}'({\mathfrak n}_{\mathbb R}^+)\) characterizes non-local symmetric breaking operators under the assumption that \(P'N^-P=G\) (Theorem 3.4). On rank one orthogonal groups, by using F-method, the author shows that Kummer’s relation on Gauss hypergeometric functions \({}_2F_1\) corresponds to functional equations for symmetric breaking operators, and moreover, the fact that the Gegenbauer polynomial of even degree is given by \({}_2F_1\) corresponds to a residue formula for regular symmetric breaking operators (Theorems 5.6 and 5.2).

22E46 Semisimple Lie groups and their representations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
53C35 Differential geometry of symmetric spaces
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