Tate Cohomology of Whittaker Lattices and Base Change of Generic Representations of GLn

Let p and l be two distinct odd primes, and let n≥ 2 be a positive integer. Let E be a finite Galois extension of degree l of a p-adic field F. Let q be the cardinality of the residue field of F. Let π F be an integral l-adic generic representation of GLₙ(F), and let π E be the base change of π F. Let Jₗ(π F) (resp. Jₗ(π E)) be the unique generic component of the mod-l reduction rₗ(π F) (resp. rₗ(π E)). Assuming that l does not divide |GLₙ₋₁(Fq)|, we prove that the Frobenius twist of Jₗ(π F) is the unique generic subquotient of the Tate cohomology group H⁰(Gal(E/F), Jₗ(π E))—considered as a representation of GLₙ(F).