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Since submitting this, I have come across a paper by M. E. Novodvorskii, “On certain invariant vectors of infinite dimensional representations of Chevalley groups”, Functional Analysis and Its Applications, Vol. 5 (1971), pp. 87–88 (in Russian), which contains my Theorem 1 below. The proof he indicates is much simpler than mine, but not exhibiting any relationship to the zeta function.
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Casselman, W. On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973). https://doi.org/10.1007/BF01428197
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DOI: https://doi.org/10.1007/BF01428197