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I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).

Suppose we have a group $G$ and subgroups $A$ and $B$ such that $A \cap B = \{1\}$ and for every $g \in G$, there exists $a \in A, b \in B$ such that $g = ab$. Then we can express $G = A \bowtie B$ as a Zappa–Szép product. This of course reduces to the semidirect or direct product in the nice cases.

Then suppose, we have sufficiently nice representations of $A$ on an $F$-vector space V, and $B$ on an $F$-vector space W, then can we find a representation of $G$ which in some sense preserves the representations of $A$ and $B$?

I've been told that the solution for semidirect products uses something called Clifford Theory, but we don't have a semidirect product here.

Our problem involves a monoid, not a group, but the Zappa-Szep product is constructed the same way there.

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Here is one way to make this question precise: there are restriction functors Rep(G) -> Rep(A) and Rep(G) -> Rep(B) which together give a functor Rep(G) -> Rep(A) x Rep(B). Does this functor have a left adjoint (which would be some kind of generalized induction)? – Qiaochu Yuan Jul 30 '10 at 7:21
The left adjoint to a product of two functors is the coproduct of the left adjoints (assuming everything exists). So the desired functor is Ind$_A^G$ $\oplus$ Ind$_B^G$. – Kevin Ventullo Jul 30 '10 at 10:27

It is hard to prove that there is no "nice" relationship between the representations of $A$ and $B$ on one hand and $G$ on the other, but experience with representation theory suggests that no nice relationship exists. For example, $A$ and $B$ can have nontrivial one-dimensional representations while $G$ has no such representation.

Even in the case where $A$ is normal, so we have a semidirect product, there is not really a clear relationship. In that case, each irreducible of $G$ is associated with a conjugacy class of representations of $A$, but there is no connection in general with representations of $B$, but only with appropriate "projective representations" of certain subgroups of $B$.

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