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Question

Let $G$ be a p-adic group, and use $\mathcal{H}(G)$ to denote the space of locally constant compactly supported functions on $G$. $\mathcal{H}(G)$ becomes an associative algebra under convolution. We know that nondegenerate modules of $\mathcal{H}(G)$ corresponds to smooth representations of $G$.

Now let $S$ be a right $\mathcal{H}(G)$-module, and $V$ be a left $\mathcal{H}(G)$-module, both are nondegenerate. Consider the tensor product $S\otimes V$. Let $N_1$ to be the subspace of $S\otimes V$ spanned by the elements of the form $$s.f\otimes v- s\otimes f.v, \forall f\in \mathcal{H}(G), s\in S, v\in V.$$ Define $$S\otimes_{\mathcal{H}(G)} V := (S\otimes V)/N_1.$$

On the other hand, we can view both $S$ and $V$ as $G$-representations, one acts from right, and the other from left. Let $N_2$ be the subspace of $S\otimes V$ spanned by the elements of the form $$s.g\otimes v- s\otimes g.v, \forall g\in G, s\in S, v\in V. $$ Take the quotient $$S\otimes_G V:= (S\otimes V)/N_2.$$

Now I want to show that $$S\otimes_{\mathcal{H}(G)} V= S\otimes_G V$$ by showing that $N_1=N_2$. I can show the inclusion $N_1 \subset N_2$, but the converse direction is not clear.

I'm wondering can we expect the above equality and how to prove it?

On the other hand, for any element $g\in G$, if we can find some compact open subgroup $K$, depending on $g$, such that $Kg=gK$, then one can deduce the above equality. But this seems open to me, and I don't know if one can expect this.

Answer 1

I claim $N_2\subset N_1$. Indeed, for any $s\in S$, $v\in V$, and $g\in G$, there exists a compact open $K$ such that $s*e_K=s$ and $e_K*v=v$. $$\begin{align*} sg\otimes v-s\otimes gv&=(sg\otimes v-s\otimes(e_{Kg}*v))+(s\otimes(e_{Kg}*v)-s\otimes gv)\\ &=\big(s*e_{Kg}\otimes v-s\otimes (e_{Kg}*v)\big)+\big(s\otimes (e_{K}*gv)-(s*e_K)\big)\\ &\in N_1. \end{align*}$$ Conversely, I claim $N_1\subset N_2$. Indeed, for any $s\in S$, $v\in V$, and $f\in\mathcal H(G)$, we hope to show $$sf\otimes v-s\otimes fv\in N_2.$$ Lemma For any compact open $K\subset G$, we have $se_K\otimes v-s\otimes e_Kv\in N_2$.

Proof Choose a normal open subgroup $K'\subset K$ such that $s*e_{K'}=s$ and $e_{K'}*v=v$. Then $$ \begin{align*} se_{K}\otimes v-s\otimes e_Kv&=\frac1{|K/K'|}\sum_{h\in G'/G}(se_{K'h}\otimes v-s\otimes e_{hK'}v)\\ &=\frac1{|K/K'|}\sum_{h\in G'/G}(sh\otimes v-s\otimes hv)\\ &\in N_1. \end{align*}$$ QED.

For the general case assume $f=e_{KgK}$ for some $g\in G$ and compact open $K\subset G$, where $s*e_K=s$ and $e_K*v=v$. Then $$\begin{align*} se_{KgK}\otimes v-s\otimes e_{KgK}v&=(s*e_{gK})\otimes v-s\otimes (e_{Kg}*v)\\ &=(sg*e_{K})\otimes v-s\otimes (e_{K}*gv)\\ &\in N_1. \end{align*}$$