Eigenfunctions of the Weil representation of unitary groups of one variable

In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the -adic case when is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke -functions in separate papers. We also recover Moen's results on when a local theta lifting from to itself does not vanish.