Symmetry breaking operators for line bundles over real projective spaces

Jan Frahm, Clemens Weiske

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Abstract

The space of smooth sections of an equivariant line bundle over the real projective space RP n forms a natural representation of the group GL(n + 1, R) . We explicitly construct and classify all intertwining operators between such representations of GL(n + 1, R) and its subgroup GL(n, R), intertwining for the subgroup. Intertwining operators of this form are called symmetry breaking operators, and they describe the occurrence of a representation of GL(n, R) inside the restriction of a representation of GL(n + 1, R). In this way, our results contribute to the study of branching problems for the real reductive pair (GL(n + 1, R), GL(n, R)). The analogous classification is carried out for intertwining operators between algebraic sections of line bundles, where the Lie group action of GL(n, R) is replaced by the action of its Lie algebra gl(n, R), and it turns out that all intertwining operators arise as restrictions of operators between smooth sections.

Original languageEnglish
JournalJournal of Lie Theory
Volume29
Issue2
Pages (from-to)511-558
Number of pages48
ISSN0949-5932
Publication statusPublished - 2019
Externally publishedYes

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