# Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

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.

No proof was given for the existence of $$H$$. I was wondering if anyone knew a reference for this result.

The existence of such subgroups $$H$$ is proved in section 2, paragraph 2.1 b) (pages 15 and 16) of
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 $$K$$.
• 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). – LSpice Dec 13 '18 at 20:22