a product of simplices. Thus $$H_1(G,M)\cong H_1(G,  {\mathcal S}(G/TU)\oplus {\mathcal S}(G/TU'))\cong H_1(T,{{\mathbb C}})\oplus H_1(T,{{\mathbb C}}).$$ Since ${\mathcal S}(G/TU)\cong H_1(T,{{\mathbb C}})$ we see that $$CoKer\left( H_1(G, {\mathcal S}(G/T)) \to H_1(G,M) \right)\cong H_1(T,{{\mathbb C}}),$$ which is one-dimensional.

The case of $PGL(2)$
--------------------

To simplify the argument we assume in this subsection that $char(F)=0$.

\[imbedding\]Conjecture \[imb\] is true for $G=PGL(2)$.

The rest of the subsection is devoted to the proof of the Theorem.

We have ${\mathcal K}_{cusp}\subset {\mathcal K}$.

\[inj\]The map $\tau : H_0(G,{\mathcal K})\to {\mathcal D}^G$ is an embedding.

We need more notation.

-   For ${\epsilon}\geq 0$ we define $G^ {\epsilon}=\{ g\in G| |p (g)| \leq {\epsilon}\}$, where $p(g)=\frac{tr^2({\tilde}g)}{det ({\tilde}g)}-4$; here ${\tilde}g\in GL(2,F)$ is a representative of $g$.

-   We let $G_s$, $G_e$, $N$, $\bar N$ denote, respectively, the sets of regular semisimple split, regular semisimple elliptic, regular unipotent and all unipotent elements.

-   We set $${\mathcal K}_u={\mathcal K}(\bar N):=\{ f\in {\mathcal S}(\bar N)\  | \int \limits_{u\in U}f 
    (u)du =0\ \forall B=TU\subset G\},$$ $${\mathcal K}_0:=\{ f\in {\mathcal K}_u| f(e)=0\},$$ where $B$ runs over the set of Borel subgroups in $G$.

-   For $f\in {\mathcal K}$ we denote by ${\kappa}(f)\in {\mathcal K}_u$ the restriction of $f$ to $\bar N$ and by $[{\kappa}(f)]$ the 