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Let   F   be a non‐archimedean local field, and    n  1    and    n  2    two positive even integers. We prove that if    π  1    and    π  2    are two smooth representations of    G  L  (   n  1   ,  F  )    and    G  L  (   n  2   ,  F  )    respectively, both admitting a Shalika functional, then the normalised parabolically induced representationπ  1   ×   π  2also admits a Shalika functional. Combining this with our recent work, we obtain as a corollary the classification of generic representations of    G  L  (  n  ,  F  )    admitting a Shalika functional when   F   has characteristic zero. This result is relevant to the study of the Jacquet–Shalika exterior square   L   factor.

Shalika periods and parabolic induction for G L ( n ) over a non‐archimedean local field