On certain eigenvectors in discrete representations of Chevalley groups

We consider Chevalley groups over disconnected locally compact fields and subgroups of them which contain their derived groups; we prove that any representation of such a group $\widetilde G$ which is nontrivial on the derived group contains an infinite-dimensional subspace of $\nu$-eigenvectors of the subgroup $\widetilde B_\mathfrak O$, where $\widetilde B_\mathfrak O$ is the intersection of the group $\widetilde G$ with the group of integral points of a Borel subgroup, and $\nu$ is an arbitrary character of it. In passing we prove that any open subgroup of the derived group is compact and is contained in only a finite number of its subgroups.
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