*Definitions*: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.  

 Let the *fixed-point subspace* $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.  

Let the *pointwise stabilizer subgroup* $G\_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$. 


Let $G$ be a finite group, $H$ a subgroup, $U$ and $V$ two irreducible complex representations of $G$.  

Let the subgroup $L := G\_{(U^H)} \cap G\_{(V^H)}$. 


*Question*: Is there an irreducible $W \le U \otimes V$ such that $G\_{(W^H)} \cap G\_{(V^H)}$ and $G\_{(U^H)} \cap G\_{(W^H)} \subset L$? 



A problem with pointwise stabilizer subgroups of fixed-point subspaces I
Let $G=C\_2\times C\_2$ and $H$ a subgroup of order $2$.


Let $U$ be the non-trivial irreducible module on which $H$ acts trivially, and $V$ one of the other non-trivial irreducible modules. Then $U\otimes V$ is irreducible, so $W$ can only be $U\otimes V$.


Then $G\_{(U^H)}=H$, and $G\_{(V^H)}=G\_{(W^H)}=G$, so $G\_{(W^H)}\cap G\_{(V^H)}\not\subset G\_{(U^H)}\cap G\_{(V^H)}$.


