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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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References:
 [1] Atiyah, M.: Resolution of singularities and division of distributions, Commun. pure appl. Math. 23, 145-150 (1970) · Zbl 0188.19405 · doi:10.1002/cpa.3160230202 [2] Bernstein, I. N.; Gelfand, S. I.: Meromorphic property of the functions $$P{\lambda}$$, Funkc. anal. Prilozh. 3, 84-85 (1969) · Zbl 0208.15201 · doi:10.1007/BF01078276 [3] Binegar, B.; Zierau, R.: Unitarization of a singular representation of $$SO(p,q)$$, Commun. math. Phys. 138, 245-258 (1991) · Zbl 0748.22009 · doi:10.1007/BF02099491 [4] Clare, P.: On the degenerate principal series of complex symplectic groups, J. funct. Anal. 262, 4160-4180 (2012) · Zbl 1242.22017 [5] Clerc, J. -L.; Kobayashi, T.; ørsted, B.; Pevzner, M.: Generalized Bernstein-reznikov integrals, Math. ann. 349, 395-431 (2011) · Zbl 1207.42021 · doi:10.1007/s00208-010-0516-4 [6] Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters, Math. ann. 217, 271-285 (1975) · Zbl 0311.10030 · doi:10.1007/BF01436180 [7] Cohen, P. B.; Manin, Y.; Zagier, D.: Automorphic pseudodifferential operators, Prog. nonlinear differ. Equ. appl. 26, 17-47 (1997) · Zbl 1055.11514 [8] Eastwood, M.; Graham, R.: Invariants of conformal densities, Duke math. J. 63, 633-671 (1991) · Zbl 0745.53007 · doi:10.1215/S0012-7094-91-06327-1 [9] Gelfand, I. M.; Graev, M. I.; Vilenkin, N. Ya.: Generalized functions, vol. 5: integral geometry and representation theory, (1966) · Zbl 0144.17202 [10] Gradshteyn, I. S.; Ryzhik, I. M.: Table of integrals, series, and products, (2007) · Zbl 1208.65001 [11] Gross, B.; Prasad, D.: On the decomposition of a representations of son when restricted to son-1, Can. J. Math. 44, 974-1002 (1992) · Zbl 0787.22018 · doi:10.4153/CJM-1992-060-8 [12] Harris, M.; Jakobsen, H. P.: Singular holomorphic representations and singular modular forms, Math. ann. 259, 227-244 (1982) · Zbl 0466.32017 · doi:10.1007/BF01457310 [13] Huang, J. -S.; Zhu, C. -B.: On certain small representations of indefinite orthogonal groups, Represent. theory 1, 190-206 (1997) · Zbl 0887.22016 · doi:10.1090/S1088-4165-97-00031-9 [14] Juhl, A.: Families of conformally covariant differential operators, Q-curvature and holography, Prog. math. 275 (2009) · Zbl 1177.53001 [15] Kashiwara, M.; Oshima, T.: Systems of differential equations with regular singularities and their boundary value problems, Ann. math. 106, 145-200 (1977) · Zbl 0358.35073 · doi:10.2307/1971163 [16] Kobayashi, T.: Discrete decomposability of the restriction of aq($$\lambda$$) with respect to reductive subgroups and its applications, part I, Invent. math. 117, 181-205 (1994) · Zbl 0826.22015 · doi:10.1007/BF01232239 [17] Kobayashi, T.: Introduction to harmonic analysis on real spherical homogeneous spaces, Proceedings of the 3rd summer school on number theory ”homogeneous spaces and automorphic forms”, 22-41 (1995) [18] Kobayashi, T.: Conformal geometry and global solutions to the yamabe equations on classical pseudo-Riemannian manifolds, Rend. circ. Mat. Palermo suppl. 71, No. 2, 15-40 (2003) · Zbl 1074.53031 [19] Kobayashi, T.: Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, Prog. math. 255, 45-109 (2008) · Zbl 1304.22013 [20] Kobayashi, T.: Branching problems of zuckerman derived functor modules, Contemp. math. 557, 23-40 (2011) · Zbl 1236.22006 [21] Kobayashi, T.: Restrictions of generalized Verma modules to symmetric pairs, Transform. groups 17, 523-546 (2012) · Zbl 1257.22014 [22] Kobayashi, T.: F-method for constructing equivariant differential operators, Contemp. math. 598, 141-148 (2013) [23] Kobayashi, T.: Propagation of multiplicity-freeness property for holomorphic vector bundles, Prog. math. 306, 113-140 (2013) · Zbl 1284.32011 [24] Kobayashi, T.; Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group $$O(p,q)$$, Mem. am. Math. soc. 212, No. 1000 (2011) · Zbl 1225.22001 [25] T. Kobayashi, T. Matsuki, Classification of multiplicity finite symmetric pairs, preprint. · Zbl 1298.22015 [26] Kobayashi, T.; ørsted, B.: Analysis on the minimal representation of $$O(p,q)$$. Part I, Adv. math. 180, 486-512 (2003) · Zbl 1046.22004 [27] Kobayashi, T.; ørsted, B.; Pevzner, M.: Geometric analysis on small unitary representations of $$GL(n,R)$$, J. funct. Anal. 260, 1682-1720 (2011) · Zbl 1217.22003 · doi:10.1016/j.jfa.2010.12.008 [28] Kobayashi, T.; ørsted, B.; Somberg, P.; Souček, V.: Branching laws for Verma modules and applications in parabolic geometry, part I · Zbl 1327.53044 [29] Kobayashi, T.; Oshima, T.: Finite multiplicity theorems for induction and restriction, Adv. math. 248, 921-944 (2013) · Zbl 1317.22010 [30] Kobayashi, T.; Pevzner, M.: Rankin-Cohen operators for symmetric pairs · Zbl 1342.22029 [31] T. Kobayashi, B. Speh, Symmetry breaking for representations of rank one orthogonal groups, preprint, arXiv:1310.3213. · Zbl 1283.22005 [32] Kostant, B.: The vanishing of scalar curvature and the minimal representation of $$SO(4,4)$$, Prog. math. 92, 85-124 (1990) · Zbl 0739.22012 [33] Krämer, M.: Multiplicity free subgroups of compact connected Lie groups, Arch. math. 27, 28-36 (1976) · Zbl 0322.22011 · doi:10.1007/BF01224637 [34] Matumoto, H.: On the homomorphisms between scalar generalized Verma modules · Zbl 0793.17002 [35] Rankin, R. A.: The construction of automorphic forms from the derivatives of a given form, J. indian math. Soc. 20, 103-116 (1956) · Zbl 0072.08601 [36] Sun, B.; Zhu, C. -B.: Multiplicity one theorems: the Archimedean case, Ann. math. 175, 23-44 (2012) · Zbl 1239.22014
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