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We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exteriorsquare gamma factors for irreducible cuspidal representations of$\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors areexpressed in terms of Bessel functions, or in terms of the regular charactersassociated with the cuspidal representations. We also relate our exteriorsquare gamma factors over finite fields to those over local fields throughlevel zero representations.

Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: finite field analogs and level zero representations