a well defined map $H_0(G,{{\mathcal H}}_{cusp})\to {\mathcal D}^G_{cusp} $ given by $f\mapsto \int\limits_{G} \ \frac{{^gf}}{dg} dg$ for a Haar measure $dg$ on $G$. In this section we define a larger subspace ${\mathcal K}$ in ${{\mathcal H}}$ on which the averaging map is still well defined and conjecture that the map $\tau$ defines an embedding $H_0(G,{\mathcal K}){\hookrightarrow}{\mathcal D}^G$. Moreover, we propose a conjectural description of the image of $\tau$. We prove this conjecture in the case when $G=PGL(2)$.

The conjecture
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We start with some notation. Let $ {\mathcal O}\subset F$ be the ring of integers and $\pi $ be a generator of the maximal ideal ${\mathfrak m}$ of ${\mathcal O}$. Let $val :F^\star \to {{\mathbb Z}}$ be the valuation such that $val(\pi)=1$. We define $\| x\| =q^{-val (x)}, x\in F^\star$ where $q=|{\mathcal O}/ {\mathfrak m}|$.

For any smooth $F$-variety $X$ we denote by ${\mathcal S}(X)$ the space of locally constant measures on $X$ with compact support. In the case when $X$ is a homogeneous $G$-variety with a $G$-invariant measure $dx$ the map $f \to f /dx$ identifies ${\mathcal S}(X)$ with the space of compactly supported locally constant functions on $X$. In this case we will not distinguish between functions and 