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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).

##### MSC:
 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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