Elementary spherical functions on the group $SL(2,P)$ over a field $P$, which is not locally compact, with respect to the subgroup of matrices with integral elements

It is proved that in the space $H$ of an irreducible unitary (in the $\Pi_1$-metric) representation $T(g)$ of the group $G=SL(2,P)$ over a normed field $P$ that is not locally compact there exists a vector $y_0$ satisfying the condition $T(g)y_0=y_0$, where $g$ runs over the subgroup $G_0$ of matrices $g\in G$ with integral elements. The function $(T(g)y_0,y_0)$ is calculated; also investigated are the unitary representations of $G$ containing the identity representation $G_0$.