Given a finite group $G$ and subgroups $H$ and $K$, and representation $\sigma: H \rightarrow GL(V_\sigma), \qquad \pi: K \rightarrow GL(V_\pi).$ Now consider the space of functions $f: G \rightarrow Hom_{\mathbb{C}}(V_\sigma, V_\pi)$ with $f(kgh) =\pi(h) f(g)\sigma(h)$ is isomorphic to the space of intertwiner $Hom_G( Ind^G_K \pi, Ind^G_H \sigma)$. If $\pi =\sigma$, it is actually a isomorphism of algebras. Similarly, we have $Res_K Ind_H^G \pi = \bigoplus_{H \gamma K \in H \backslash G /K} Ind_{H^\gamma \cap K}^H Res_{H^\gamma \cap K} \pi^\gamma.$ These essentially carry over to $G$ compact.
To which category of representation of locally compact groups or algebraic groups, do we have natural generalizations of these theorems? What kind of generalized functions do we have to consider $f : G \rightarrow Hom(-,-)$ do we have to consider: functions, distributions, measures? (=> Schwartz kernel theorem useful?) What measure on $K \in H \backslash G /K$?