This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected.
It seems that the restriction functor of representations of Lie groups $H \hookrightarrow G$ has a left adjoint, induction, and a right adjoint, coinduction. When are these functors isomorphic?
My guess was compact Lie groups, but I couldn't find anything on that. (It's true in 0 dimensions. Haha.) Then in Kirillov's book "Elements of the theory of representations", I find the statement that they are isomorphic for semisimple algebras, which made me think of the possibility that it might be true for semisimple Lie groups/algebras.