2025/07/13 by Sabyasachi Dhar, Dhar, Sabyasachi, Santosh Nadimpalli +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2507.09773
openalex publication_date 2025/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected reductive group defined over a non-Archimedean local field F of residue characteristic p. Let ℓ be a prime number distinct from p. Let E be a cyclic Galois extension of F with [E:F]=ℓ. Let Π be a finite length \mathbbF_ℓ-representation (or an ℓ-modular representation) of G(E)\rtimes \rm Gal(E/F). In this context, we prove a conjecture of Treumann and Venkatesh which predicts that the Tate cohomology groups \widehatHi(\rm Gal(E/F), Π) are finite length representations of G(F). We discuss the explicit computation of these Tate cohomology groups when G is \rm GLn and Π is obtained as a base change lifting of a depth-zero cuspidal representation of \rm 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 \rm Gal(\mathbbFq^ℓ/\mathbbFq)-Tate cohomology groups of the mod-ℓ reduction of the unipotent cuspidal representation of \rm Sp4(\mathbbFq^ℓ).