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Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ is the Hecke algebra of $G$. That is, $$ {\rm Rep}(G)\simeq \mathcal{H}(G){\text -}{\rm mod}. $$ Now let us consider the compactly induction functor $$ c{\text -}{\rm Ind}_{H}^{G}\colon {\rm Rep}(H)\longrightarrow {\rm Rep}(G). $$ For any object $(\sigma,W)$ of ${\rm Rep}(H)$, we have $c{\text -}{\rm Ind}_{H}^{G}\sigma =(\Sigma,X_{c}).$ The space $X_{c}$ is $$ X_{c}=\{f\colon G\longrightarrow W\mathrel{\vert} (1),(2),(3)\}, $$

  • (1). for any $h\in H,g\in G$, $f(hg)=\sigma(h)f(g),$
  • (2). there exists a open compact subgroup $K_{f}$ such that $f(gk)=f(g)$ for any $k\in K_{f}$
  • (3). the support of $f$, ${\rm supp}(f)$ is compact modulo $H$.

And $\Sigma$ is $$ \Sigma\colon G\longrightarrow {\rm Aut}(X_{c});g\longmapsto \left(f\longmapsto (x\longmapsto f(xg))\right). $$

I want to know about the functor $c{\text -}{\rm Ind}_{H}^{G}\colon \mathcal{H}(H){\text -}{\rm mod}\longrightarrow \mathcal{H}(G){\text -}{\rm mod}$. Probably this is given as a thensor product. Let $\mu_{G}$ and $\mu_{H}$ be left Haar measures on $G,H$ respectively. For any $\phi\in \mathcal{H}(H)$ and $f\in \mathcal{H}(G)$, we define $R(\phi)f\in \mathcal{H}(G)$ to be $$ g\longmapsto \int_{H}\phi(h)f(hg)d\mu_{H}(h). $$ Then $R$ provides $\mathcal{H}(G)$ with a structure of right smooth module over $\mathcal{H}(H)$. Perhaps there exists a canonical ismorphism $c{\text -}{\rm Ind}_{H}^{G}\sigma \simeq \mathcal{H}(G)\otimes_{\mathcal{H}(H)}\sigma$. But I dont know the explicit correspondence of this isomorphism.

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  • $\begingroup$ You will also need the notion of sufficiently many idempotents in a ring (such as these full Hecke algebras), and correct formation of tensor products over them. And pay attention to the modular functions (regarding left-versus-right Haar measures...) $\endgroup$ Apr 23, 2021 at 18:17

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