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Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup of $G$ in good position relative to $P_0$, so we have $G = P_0K$ (if $G$ is unramified, we can take $K$ to be a conjugate of $G(\mathcal O_k)$).

For each standard parabolic $P$, let $M$ be the unique Levi subgroup of $P$ containing $A_0$.

Does there always exist an open subgroup $H$ of $K$ such that we have $$H = \bigg(\prod\limits_{\alpha \in \Phi(A_M,P)_{\textrm{red}}}(H \cap N_{-\alpha})\bigg)(H \cap M)\bigg(\prod\limits_{\alpha \in \Phi(A_M,P)_{\textrm{red}}}(H \cap N_{\alpha})\bigg)$$? Here $A_M$ is the split component of $M$, and $N_{\alpha}$ is the product of the root subgroups $U_a$ for $a \in \Phi(A_0,G), a|_{A_M} = \alpha$.

This is claimed in Waldspurger's notes on the Plancherel measure for $p$-adic groups, based on Harish-Chandra's lectures, Section I.1.
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No proof was given for the existence of $H$. I was wondering if anyone knew a reference for this result.

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The existence of such subgroups $H$ is proved in section 2, paragraph 2.1 b) (pages 15 and 16) of

Bernstein, J. N. Le "centre'' de Bernstein. (French) [The Bernstein "center''] Edited by P. Deligne. Travaux en Cours, Representations of reductive groups over a local field, 1–32, Hermann, Paris, 1984.

More precisely Bernstein proves that there are arbitrary small compact open subgroups $H$ of $G$ satisfying the product decompositions (i.e. the so called Iwahori decompositions). In particular you can find one in $K$.

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    $\begingroup$ In modern language, the Moy–Prasad subgroups $G_{x, r}$ with $r > 0$, and $x$ lying in the image of the building of $M$, have this property (and form a neighbourhood base at the identity). $\endgroup$ – LSpice Dec 13 '18 at 20:22

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