Middle supercuspidals of GL(2N,F) are characterized by twisted gamma factors against tamely ramified quasi-characters and simple supercuspidals of GL(N,F).
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On the Local Converse Theorem for Depth $\frac{1}{N}$ Supercuspidal Representations of $\text{GL}(2N, F)$
Middle supercuspidals of GL(2N,F) are characterized by twisted gamma factors against tamely ramified quasi-characters and simple supercuspidals of GL(N,F).