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Let $G$ be $p$-adic group and let $G \rightarrow GL(V)$ be a representation. For example, $V$ is a quadratic $\mathbb{Q}_p$-space and $G$ is the associated orthogonal group.

Take a point $v \in V$, let $H \subset G$ be its stabilizer and assume $X = H \backslash G = G \cdot v \subset V$ is a closed subvariety. To continue with the example, $H = O(v^{\perp})$.

I consider the indicator function of (the image via the projection $G \rightarrow H \backslash G$ of) $H \cdot K$, for some compact open subgroup $K$, and I want to extend this indicator function to a locally constant smooth function on $V$.

Is it somehow possible ? Do you know of any work that addresses similar questions ?

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    $\begingroup$ It's a bit unclear to me when you are working with rational points, and when not. For example, do you want to regard $X$ as $G(\mathbb Q_p)\cdot v$ or $(G\cdot v)(\mathbb Q_p)$, which can be larger? Is your $p$-adic field always of characteristic $0$, or might it be, say, $\mathbb F_p((t))$? But, most importantly, regardless of these subtleties, I don't understand the question: by almost any interpretation, $H\backslash H\cdot K$ is compact open in the analytic topology, so its indicator is already locally constant (which, to me, in this context is what smooth means). What is missing? $\endgroup$
    – LSpice
    Commented Jul 4 at 13:57
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    $\begingroup$ Yes sorry I am working with fields of characteristic $0$ and I meant $(G \cdot v)(\mathbb{Q}_p)$. I'm looking for a locally constant and smooth function $f : V \rightarrow \mathbb{C}$ whose restriction to $G \cdot v$ (which may be viewed as a function on $H \backslash G$) is the indicator function of $H \backslash H \cdot K$. $\endgroup$
    – Sentem
    Commented Jul 4 at 14:29

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Since $H^1(\mathbb Q_p, H)$ is finite and each fibre of $(G\cdot v)(\mathbb Q_p) \to H^1(\mathbb Q_p, H)$ is closed, we have that the fibre $G(\mathbb Q_p)\cdot v$ over the trivial cohomology class is open (as well as closed). Therefore, $G(\mathbb Q_p) \to (G\cdot v)(\mathbb Q_p)$ is an open map, so $K\cdot v$ is an analytically open subset of $(G\cdot v)(\mathbb Q_p)$.

Since $G\cdot v$ is a closed subvariety of $V$, we have that the analytic topology on $(G\cdot v)(\mathbb Q_p)$ is the subspace topology coming from the analytic topology on $V(\mathbb Q_p)$. Thus $K\cdot v$ is of the form $U \cap (G\cdot v)(\mathbb Q_p)$ for some analytically open subset $U$ of $V(\mathbb Q_p)$. Since the compact, open subsets of $V(\mathbb Q_p)$ form a basis for its topology, we can write $U$ as a union of compact, open subsets of $V(\mathbb Q_p)$. Since $K\cdot v$ is compact, finitely many of these compact, open subsets of $V$ suffice to cover it. Their union is compact, open, and still intersects $(G\cdot v)(\mathbb Q_p)$ in $K\cdot v$, so its indicator function is a smooth extension of the indicator function of $K\cdot v$.

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