Natural Differential Operators in Parabolic Geometry and Branching Problems. Symposium on Representation Theory 2012. Kagoshima, Japan, 4-7 December 2012.

As an analogy of unitary representation without continuous spectrum (“discrete decomposable representations”), we give a geometric criterion for the property “having non-trivial subrepresentations” in the restriction of Verma modules with respect to reductive symmetric pairs. As its application, I discuss intertwining operators in parabolic geometry, and in particular, propose a new method (F-method) to produce naturally Juhl's conformally equivariant differential operators and Cohen-Rankin operators in holomorphic automorphic forms, together with their generalizations.

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