This is in a sense a follow up on the popular question [Induction and Coinduction of Representations](https://mathoverflow.net/questions/1534/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.



When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
The inclusion $H \to G$ induces a map of classifying stacks $f \colon BH \to BG$ (note that these are *stacks*, not the "topologists $BG$"). As remarked in the question you linked to, induction and coinduction correspond to the functors $f\_\ast$ and $f\_!$ respectively. These functors will be equal if $f$ is proper. The fiber of $f$ is the quotient $G/H$, so a sufficient condition is that $H$ is cocompact in $G$.


