On Frobenius reciprocity theorem The classical Frobenius reciprocity theorem asserts the following:
If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res U})_{H}.$$
The proof in the standard textbook (Fulton&Harris, Dummit&Foote,etc) is easy to understand. What puzzled me is this Frobenius theorem that appears in Raoul Bott's paper:
"Proposition 2.1. Let $W$ be a $G$-module, let $M$ be an $H$-module and denote by $i^{*}W$, the restriction of $W$ to $H$. Then, $$Hom_{G}(W,\Gamma MG)\cong Hom_{H}(i^{*}W, M).$$
In here the $\Gamma MG$ is defined to be the section of the bundle $G\times_{H} M\rightarrow G/H$, with $G\times_{H}M$ defined to be $G\times M/(g,m)\approx (gh,h^{-1}m)$. 
Bott claimed in his paper (Homogeneous differential operators) that this isomorphism is quite canoical, yet not only I could not understand his proof, but also I could not see how the isomorphism is anything but canonical. There should be some kind of relationship between this and the classical theorem, but I could not get it as well. After several hours pondering I decided to ask in here as the matter is purely technical. 
 A: Sections of the bundle are the same as maps $\phi:G \to M$ such that $\phi(gh) = h^{-1}\phi(g)$ for all $g \in G, h \in H$.  This is one way to define the induction of $M$ from $H$ to $G$ in the context of not-necessarily finite groups.
Now the isomorphism you ask about is the adjunction
$$Hom_G(W,Ind_H^G M) \cong Hom_H(W,M).$$
The map is easy to define: map $Ind_H^G M \to M$ via evaluation at $1 \in G$
(in Bott's geometric terms, look at the value of the section over $g =1$) --- this is $H$-equivariant, and induces a corresponding map of Hom spaces.  To check it
is an isomorphism is not much harder.  (If sections/maps are understood to be smooth, then you will have to use the fact that $W$ is a smooth representation
of $G$.)

Aside: you shouldn't be surprised at the appearance of the term "adjunction" here; that is what Bott's formula is, and all the basic facts from character theory of finite groups are manifestations of underlying facts about the categories of representations.  Also, it is easier for many people (including me!) to prove the categorical facts first and then to deduce the character-theoretic facts directly, since there is more structure to work with.
