Geodesic
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
Izvestiya. Mathematics , Tome 1 (1967) no. 2, pp. 349-380.

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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 $G_0$. 
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R. S. Ismagilov. 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. Izvestiya. Mathematics , Tome 1 (1967) no. 2, pp. 349-380. http://geodesic.mathdoc.fr/item/IM2_1967_1_2_a9/

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