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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.

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    $\begingroup$ Small edit: your "coadjoint" in line 4 is meant to be "coinduction". $\endgroup$ – Jim Humphreys Sep 10 '14 at 19:02
  • $\begingroup$ A bit late to the party, but I add a link here to a related question of mine, also from 2014: mathoverflow.net/q/188571/41139. I think many of the references in the comments, answer and question itself have relevance for the question stated here. But besides that, the answer below stating that induction and coinduction are equal if $H$ is co-compact in $G$ seems to be at odds with the situation in that question where the two types of induction are not equal but the group $H$ (the minimal parabolic subgroup) is co-compact by the Iwasawa decomposition. $\endgroup$ – Vincent Aug 29 '17 at 12:22
  • $\begingroup$ I will try and understand this seeming discrepancy better, but later. So a link is useful in finding the two questions again. (For instance I didn't see the current question when typing the linked question) $\endgroup$ – Vincent Aug 29 '17 at 12:23
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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 $G$.

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  • $\begingroup$ Awesome, so compact homogeneous spaces are an example! $\endgroup$ – Manuel Bärenz Sep 10 '14 at 14:22
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    $\begingroup$ @Sasha, if we're doing representation theory of Lie groups, were does a choice of sheaf category ever enter? In the choice of what kind of stack to consider? $\endgroup$ – Manuel Bärenz Sep 11 '14 at 9:01
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    $\begingroup$ @Sasha, I meant representations are representations and you can define them without ever referring to constructible sheaves or coherent sheaves, so the answer to the question should be independent of that, right? $\endgroup$ – Manuel Bärenz Sep 11 '14 at 10:10
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    $\begingroup$ @Turion: Of course. What I mean is that Dan in his answer implicitely identifies the category of representations with the category of sheaves on $BG$. But since he does not mention which sheaves are those, it is not clear which condition you really need. $\endgroup$ – Sasha Sep 11 '14 at 12:22
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    $\begingroup$ @Sasha, I see, thanks. Dan, which sheaves do you mean then? I couldn't find any material that spells out this correspondence, but since I know close to nothing about stacks and sheaves, that's no surprise. Maybe you know an easier introduction or review where I could find that out? $\endgroup$ – Manuel Bärenz Sep 11 '14 at 16:28

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