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We consider several questions about restriction of representations ofclassical and metaplectic groups over local and global fields to subgroups,extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. Thisincludes Bessel and Fourier-Jacobi models too. We formulate several conjecturesabout these restriction problems involving root numbers of symplecticrepresentations in the local case, and central critical L-value in the globalcase. Along the way we prove several results both in number theory andrepresentation theory.

Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups