Titlebook: Symmetry Breaking for Representations of Rank One Orthogonal Groups II; Toshiyuki Kobayashi,Birgit Speh Book 2018 Springer Nature Singapor

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,Minor Summation Formulæ Related to Exterior Tensor ,,This chapter collects some combinatorial formulæ, which will be used in later chapters to compute the (., .)-spectrum for symmetry breaking operators between differential forms on spheres .. and .., namely, between principal series representations ..(., .) of . and ..(., .) of its subgroup . in the setting where ..

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,The Knapp–Stein Intertwining Operators Revisited: Renormalization and ,-spectrum,In this chapter, we discuss the classical Knapp–Stein operators, which may be viewed as a baby case of symmetry breaking operators (., . = . case).

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Symmetry Breaking Operators for Irreducible Representations with Infinitesimal Character ,: Proof oIn the first half of this chapter, we give a proof of Theorems . and . that determine the dimension of the space of symmetry breaking operators from . representations Π of . = .(. + 1, 1) to . representations . of the subgroup . = .(., 1) when both Π and . have the trivial infinitesimal characters ., or equivalently by Theorem . (2).

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Application I: Some Conjectures by B. Gross and D. Prasad: Restrictions of Tempered RepresentationsInspired by automorphic forms and .-functions, B. Gross and D. Prasad published in 1992 conjectured about the restriction of irreducibletempered representations of special orthogonal groups .(. + 1, .) to a special orthogonal subgroup .(., .).

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A Conjecture: Symmetry Breaking for Irreducible Representations with Regular Integral InfinitesimalWe conjecture that Theorems . and . hold in more generality. We will formalize and explain this conjecture in this chapter more precisely and provide some supporting evidence.

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,Appendix I: Irreducible Representations of , = ,(, + 1, 1), ,-stable Parameters, and Cohomological In Appendix I, we give a classification of irreducible admissible representations of . = .(. + 1, 1) in Theorem 14.36.

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Toshiyuki Kobayashi,Birgit SpehIntroduces a new method to construct and classify matrix-valued symmetry breaking operators in representation theory.Includes hot topics of automorphic forms and conformal geometry as applications of