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Abstract 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 ₅ be an integral l-adic generic representation of GL₍ (F), and let ₄ be the base change of ₅. Let J₋ (₅) (resp. J₋ (₄) ) be the unique generic component of the mod-l reduction r₋ (₅) (resp. r₋ (₄) ). Assuming that l does not divide |GL₍-₁ (Fₐ) |, we prove that the Frobenius twist of J₋ (₅) is the unique generic subquotient of the Tate cohomology group H^0 (Gal (E/F), J₋ (₄) ) —considered as a representation of GL₍ (F).
Nadimpalli et al. (Fri,) studied this question.