Let $G$ be a locally profinite group, and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth representation of $G$, and let $\tau$ be a smooth representation of $H$ (henceforth, every representation is assumed to be over the complex field). Is there a $G$-isomorphism
$$ \sigma \otimes\operatorname{c-Ind}_H^G(\tau) \cong \operatorname{c-Ind}_H^G(\sigma|_H \otimes \tau), $$
where $\operatorname{c-Ind}_H^G(-)$ denotes compact induction from $H$ to $G$, and $\sigma|_H$ is the restriction of $\sigma$ to $H$?
The above isomorphism is known when $G$ and $H$ are both finite groups, and both $\sigma$ and $\tau$ are finite-dimensional representations (in this case, $\operatorname{c-Ind}_H^G(-)$ is naturally isomorphic to extension by scalars $(-)_H^G := \mathbb{C}G \otimes_{\mathbb{C}H}(-)$). This can be seen, for example, as a corollary of the following instance of Mackey's tensor product theorem:
Let $G$ be a finite group, let $M,N$ be subgroups of $G$, and suppose that $G=MN$. If $\sigma$ and $\tau$ are finite-dimensional representations of $M$ and $N$ respectively, then there is a $G$-isomorphism: $$ \sigma_M^G \otimes \tau_N^G \cong \left(\; (\sigma \otimes \tau )|_{M \cap N} \; \right)_{M \cap N}^G $$
So in particular, the desired formula is obtained from the above result (for finite-dimensional representations of finite groups) if we take $G=M$ and $N=H$. However, I am afraid that an adaptation of this approach might fail in the setting of smooth representations. Indeed, I am not sure if in this case there is a $G$-isomorphism
$$ \operatorname{c-Ind}_M^G(\sigma) \otimes \operatorname{c-Ind}_N^G(\tau) \cong \operatorname{c-Ind}_{M \cap N}^G \left( \; (\sigma \otimes \tau)|_{M \cap N} \; \right)$$,
where M and N are closed subgroups of $G$ such that $G=MN$, and $\sigma, \tau$ are smooth representations of $M$ and $N$ respectively. Heuristically, taking the tensor product of smooth representations might "largely increase" the subspaces of invariants under a given compact open subgroup $K$ of $G$: in particular, $(X_\sigma \otimes_{\mathbb C}X_{\tau})^K$ might properly contain $X_{\sigma}^K \otimes_{\mathbb{ C}} X_{\tau}^K$, where $X_{\sigma}$ and $X_{\tau}$ are the representation spaces of $\operatorname{c-Ind}_M^G(\sigma)$ and $\operatorname{c-Ind}_N^G(\tau)$, respectively.
I know that the particular instance of Mackey's tensor product theorem holds when $M$ and $N$ are both open in $G$ (mainly because int this case, compact induction coincides with extension by scalars, where the group algebra construction is replaced by the Hecke algebra construction), and thus the desired formula follows exactly as above. However, the groups which I am considering are rarely open. On the other hand, the best I could achieve is the following particular case:
Let $G, H, \sigma$ and $\tau$ be as in the beginning. If both $\sigma$ and $\tau$ are one-dimensional, then there is a surjective $G$-homomorphim $$ \sigma \otimes\operatorname{c-Ind}_H^G(\tau) \twoheadrightarrow \operatorname{c-Ind}_H^G(\sigma|_H \otimes \tau) $$
Essentially, I prove first that if $M,N$ are closed subgroups of $G$ and $\sigma, \tau$ are one-dimensional smooth representations of $M$ and $N$ respectively, then there is a canonical $\mathbb{C}$-linear map
$$\Psi: X_{\sigma} \otimes_{\mathbb{C}} X_{\tau} \rightarrow X_{\sigma \boxtimes \tau}$$ $$\phi \otimes \psi \mapsto \Psi_{\phi,\psi}$$
where $(-)\boxtimes(-)$ is the outer tensor product, which affords a smooth representation of $G \times G$, $X_{\sigma \boxtimes \tau}$ is the representation space of $\operatorname{c-Ind}_{M \times N}^{G \times G}(\sigma \boxtimes \tau)$, and
$$\Psi_{\phi,\psi}:G \times G \rightarrow V_{\sigma} \otimes_{\mathbb{C}} V_{\tau}$$ $$(g,h) \mapsto \phi(g) \otimes \psi(h)$$,
where $V_{\sigma}$ and $V_{\tau}$ are the representation spaces of $\sigma$ and $\tau$, respectively. Secondly, it can be shown that the map $\Psi$ is a surjective $(G \times G)$-homomorphism intertwining $\operatorname{c-Ind}_M^G(\sigma) \boxtimes \operatorname{c-Ind}_N^G(\tau)$ and $\operatorname{c-Ind}_{M \times N}^{G \times G}(\sigma \boxtimes \tau)$. Finally, we have a $G$-isomorphism
$$ \left(\operatorname{c-Ind}_M^G(\sigma) \boxtimes \operatorname{c-Ind}_N^G(\tau) \right)|_{\Delta G} \cong \operatorname{c-Ind}_M^G(\sigma) \otimes \operatorname{c-Ind}_N^G(\tau), $$
where $\Delta G= \{(g,g): g \in G\}$ is the diagonal subgroup of $G \times G$ (and which identifies with $G$). Also, using $G=MN$ we have a $G$-isomorphism
$$ \operatorname{c-Ind}_{M \times N}^{G \times G}(\sigma \boxtimes \tau)|_{\Delta G} \cong \operatorname{c-Ind}_{M \cap N}^G\left( (\sigma \otimes \tau)|_{M \cap N} \;\right)$$
The result easily follows from the above considerations.
As for the proof that $\Psi$ is a surjective $(G \times G)$-homomorphism, one first directly shows by calculations that $\Psi$ is a $(G \times G)$-homomorphism. Then one proves that for compact open subgroups of $G \times G$ of the form $K_1 \times K_2$ with $K_i$ compact open subgroup of $G$, the map $\Psi$ induces a surjective linear map
$$(X_{\sigma} \otimes_{\mathbb {C}} X_{\tau})^{K_1 \times K_2} \twoheadrightarrow (X_{\sigma \boxtimes \tau})^{K_1 \times K_2}$$.
Now, since $G$ is locally profinite, we can take a neighborhood basis of the identity in $G \times G$ given by compact open subgroups of the form $K_1 \times K_2$ with $K_i$ compact open subgroup of $G$. This implies that both representation spaces of $\operatorname{c-Ind}_M^G(\sigma) \boxtimes \operatorname{c-Ind}_N^G(\tau)$ and $\operatorname{c-Ind}_{M \times N}^{G \times G}(\sigma \boxtimes \tau)$ are unions of subspaces of invariants under such compact open subgroups (using smoothness of both representations), and by a typical argument this gives the surjectivity of $\Psi$. It might be the case that the above surjective $\mathbb{C}$-linear map fails to be injective, thus reinforcing the idea that one loses control over the "size" of subspaces of invariants under compact open subgroups when taking tensor products.
Finally, if anyone knows that the desired formula holds (or that the outer tensor product "commutes" with compact induction; the desired formula is then just a corollary, as we have seen), I would (urgently!) ask for either an argument or a reference. Thank you in advance.
