Let G be a connected reductive group defined over a non-Archimedean local field F of residue characteristic p. Let be a prime number distinct from p. Let E be a cyclic Galois extension of F with E: F=. Let Π be a finite length F_-representation (or an -modular representation) of G (E) Gal (E/F). In this context, we prove a conjecture of Treumann and Venkatesh which predicts that the Tate cohomology groups Hⁱ (Gal (E/F), Π) are finite length representations of G (F). We discuss the explicit computation of these Tate cohomology groups when G is GLₙ and Π is obtained as a base change lifting of a depth-zero cuspidal representation of GLₙ (F). The primary novelty from our previous work is that we treat the case where Π is possibly non-cuspidal. We also study the Gal (Fₐ^/Fq) -Tate cohomology groups of the mod- reduction of the unipotent cuspidal representation of Sp₄ (Fₐ^).
Dhar et al. (Sun,) studied this question.