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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 $K \in H \backslash G /K$?

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Look at Mackey's work: Systems of imprimitivity, references on Wiki + Barut and Rączka, maybe this link and references therein. – t.b. Jul 27 '11 at 17:08
Thanks for suggesting the book of Barut and Raczka – Jul 28 '11 at 8:58
up vote 1 down vote accepted

Some more-tangible examples than the general measure-theoretic "Mackey machine":

In totally-disconnected groups (e.g., p-adic), there is a useful "Mackey-Bruhat" theory (as in papers of Bruhat in the 1950s and early 1960s), as in Bernstein-Zelevinsky's papers on $GL_n(\mathbb Q_p)$. The direct sum becomes a filtration.

For Lie groups, the fact that distributions supported on sub-objects can have normal components means that even the filtration is not correct in that case, but must be enlarged by including normal jet bundles. Decomposing $C^\infty_c(\mathbb R)$ or $L^2(\mathbb R)$ as $\mathbb R^\times$ repn space already illustrates this. Nevertheless, since the normal jet bundle tends to give "integer" shifts in repn parameters, many problems can be solved in lucky cases. A simple not-so-trivial case is the theta-pairing between $O(2)$ and/or $O(1,1)$ and $SL_2(\mathbb R)$ (taking advantage of the splitting of the metaplectic group...) acting on Schwartz functions on $\mathbb R^2$. The $O(1,1)$ case is the less-trivial of the two.

In short, yes, Schwartz kernels in general, but in nice circumstances the required distributions have pleasant homogeniety properties, so there are strong/useful constraints.

Edit: Prompted by a comment, yes the above refers to categories of "smooth" repns, with the two senses appropriate to repns of totally disconnected and Lie groups, resp.

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So, I guess you work in the category of smooth, admissible representation, where as Mackey Machine applies to unitary stuff only? – Jul 28 '11 at 9:01

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