Induction and Compact induction of representations Let $H \leq G$ be a subgroup of a finite group, $G.$  Suppose $(\sigma, W)$ is a representation of $H.$ Then we know that $Ind_H^G \sigma $ and $ind_H^G \sigma $ are isomorphic, where 
$$Ind_H^G \sigma = \{f:G \rightarrow W | f(hg) = \sigma (h) f(g) \ \  \forall h\in H, g \in G \}$$ and 
$$ind_H^G \sigma = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W.$$
My guess is that this can be generalized further where we can accommodate infinite groups and finite index subgroups with finite dimensional representations without much effort.
My question is how much more general can we go? Can the groups be algebraic groups in particular? Can we have subgroups which are not necessarily finite indexed and infinite dimensional representations? What combinations of these parameters work?
Any references will be appreciated.
Thanks for the help.
 A: $$\def\Ind{\mathop{\mathrm{Ind}}\nolimits} \def\ind{\mathop{\mathrm{ind}}\nolimits}
\def\Hom{\mathop{\mathrm{Hom}}\nolimits}$$
Let me talk about these two functors for a bit. 
The first thing I should mention is that while the representations $\Ind_{H}^{G} \sigma$ and $\ind_{H}^{G} \sigma$ are isomorphic (for finite groups $G$ and $H$), they are not necessarily canonically isomorphic. The reason that the representations turn out to be isomorphic is because the representation category of a finite group over a field of characteristic prime to its order is semisimple. What this implies in particular is that for representations $V, W$ of $G$,
$$\Hom_{G}(V, W) \cong \Hom_{G}(W, V)$$
Now, the first functor $\Ind$, is right adjoint to restriction. If $W$ is a representation of $H$ and $V$ a representation of $G$, then
$$\Hom_{H}(V, W) \cong \Hom_{G}(V, \Ind_{H}^{G} W).$$
Here the isomorphism sends $\phi : V \rightarrow W$ to $\Phi$ which sends $v$ to the function $g \mapsto \phi(g \cdot v)$. 
On the other hand, the second functor $\ind$, is left adjoint to restriction i.e.
$$\Hom_{H}(W, V) \cong \Hom_{G}(\ind_{H}^{G} W, V)$$
where $\phi: W \rightarrow V$ is sent to $\Phi$ which maps $g \otimes w$ to $g\cdot \phi(w)$.
Since $\Hom$ spaces can just be switched around in the semisimple representation category of a finite group, the two functors give isomorphic representations in this case. But the isomorphism
$$\Hom_{G}(V, W) \cong \Hom_{G}(W, V)$$
is very non-canonical. You have to choose some isomorphism and there isn't any particularly good one. So, the functors $\Ind$ and $\ind$ are not isomorphic.
I can try and explain where these functors extend and where they don't when you impose additional conditions on the groups.


*

*Infinite groups with finite index: If $G$ and $H$ are infinite groups with $H$ finite index in $G$, both functors definitely exist and make sense at the level of finite dimensional representations. These functors will probably give isomorphic representations if $\sigma$ is semisimple. However, I am not sure if the representations will be isomorphic if $\sigma$ is not semisimple.

*Algebraic Groups: For finite dimensional representations of an algebraic group, $\Ind$ makes perfect sense as long as you restrict to regular maps (and is often called geometric induction) and the field is algebraically closed. There's actually a slightly better description of this functor which works also in the non-algebraically closed setting: given a representation $H$ of $W$, we can form a vector bundle over the variety $G/H$
$$G \times_{H} W := G \times W/(gh, w) ~ (g, hw).$$
(This definition can easily be upgraded to a scheme theoretic one.) Then, we can define $\Ind_{H}^{G} W$ as the global sections of this vector bundle. This has the same adjointness property as in the finite group setting.
Now, $\ind$ as defined does not make sense for finite dimensional representations of an algebraic group. One reason is easy to see: $\mathbb{C}[G]$ (the algebraic coordinate ring) has infinite rank over $\mathbb{C}[H]$ so the tensor product
$$\mathbb{C}[G] \otimes_{\mathbb{C}[H]} W$$
is not a finite dimensional representation of $G$. But the failure of finite dimensionality is not too bad here because $\mathbb{C}[G]$ is a sum of its finite dimensional rational subrepresentations. Hence, $\ind_{H}^{G} W$ is also a sum of finite dimensional subrepresentations and the functor can thus be defined on the ind-completion of the category of rational representations. In this setting, $\ind$ satisfies the same adjointness property as in the case of finite groups. 
But now, we need to take infinite dimensional representations so even when the algebraic groups $G$ and $H$ are reductive and the representation category is semisimple, I do not think the two functors will always give isomorphic representations on the infinite dimensional representations. I am not 100% sure on this though.


*Lie groups: I believe a similar situation happens as in the algebraic case by always using $L^{2}$ functions, either in the definition of $\Ind$ or in the definition of $\mathbb{C}[G]$ and $\mathbb{C}[H]$.


Disclaimer: I do not know much about $L^{2}$-representations of Lie groups. 


*Lie algebras: The second functor $\ind$ has a similar definition for Lie algebras by replacing $\mathbb{C}[G]$ with the universal enveloping algebra $U(\mathfrak{g})$ (and doing the same for $H$). This is a very important construction when studying semisimple Lie algebras as it's used to define Verma modules. 

*Associative algebras: $\ind$ is basically extension of scalars for algebras. If you have an algebra $A$ with a subalgebra $B$ and a left $B$-module $W$, then you can define an $A$-module $\ind_{B}^{A} W : = A \otimes_{B} W$. This defines a functor which is left adjoint to restriction.
