Tate cohomology of Whittaker lattices and Base change of cuspidal representations of GLn{\rm GL}_nGLn​ (original) (raw)

Let p and l be distinct primes and let n be a positive integer. Let E be a finite Galois extension of degree l of a p-adic field F. Let π and ρ be two l-adic integral smooth cuspidal representations of GLn(E) and GLn(F) respectively such that π is obtained as base change of ρ. Then the Tate cohomology H 0 (π), as an l-modular representation of GLn(F), is well defined. In this article, we show that H 0 (π) is isomorphic to the Frobenius twist of the reduction mod-l of the representation ρ. We assume that l, p = n and l is banal for GL n−1 (F). SABYASACHI DHAR AND SANTOSH NADIMPALLI defined in the work of G.Moss and N.Matringe in [MM22]. However, the right notion of Base change over local Artinian F l-algebras is not clear to the authors and hence, we use the mild hypothesis that l is banal for GL n−1 (F). When F is a local function field, the above theorems follow from the work of T.Feng [Fen20]. T.Feng uses the constructions of Lafforgue and Genestier-Lafforgue [GL17]. N.Ronchetti also proved the above results for depth-zero cuspidal representations using compact induction model. Our methods are very different from the work of N.Ronchetti and the work of T.Feng. We rely on Rankin-Selberg integrals and lattices in Whittaker models. We do not require the explicit construction of cuspidal representations. We use the l-adic local Langlands correspondence and various properties of local ǫ and γ-factors both in l-adic and mod-l situations associated with the representations of the p-adic group and the Weil group. The machinery of local ǫ and γfactors of both l-adic and mod-l representations of GL n (F) is made available by the seminal works of D.Helm, G.Moss, N.Matringe and R.Kurinczuk (see [HM18], [Mos16], [KM21], [KM17]). We sketch the proof of the above Theorem (1.1). It is proved using Kirillov model and using some results of Vigneras on the lattice of integral functions in a Kirillov model being an invariant lattice. The discussion in this paragraph is valid for any positive integer n. Let ψ : F → Q × l be a non-trivial additive character. Let (π F , V) be a generic l-adic representation of GL n (F), in particular, V is a Q l-vector space. Let N n (F) be the group of unipotent upper triangular matrices in GL n (F). Let Θ : N n (F) → Q × l be a non-degenerate character corresponding to ψ and we let W (π F , ψ) to be the Whittaker model of π F. Similar notations for π E are followed. Let π E be the base change of π F. It is easy to note that (Lemma 2.3) W (π E , ψ) is stable under the action of Gal(E/F) on the space Ind GLn(E) Nn(E) Θ. Let π F be an integral generic l-adic representation of GL n (F), and let W 0 (π F , ψ) be the set of Z l-valued functions in W (π F , ψ). It follows from the work of Vigneras [Vig04, Theorem 2] that the subset W 0 (π F , ψ) is a GL n (F) invariant lattice. Let K(π F , ψ) be the Kirillov model of π F , and let K 0 (π F , ψ) be the set of Z l-valued functions in K(π F , ψ). Using the result [MM22, Corollary 4.3] we get that the restriction map from W 0 (π F , ψ) to K 0 (π F , ψ) is a bijection. Let π F and π E be cuspidal representations. It follows from the work of Treumann-Venkatesh that H(Gal(E/F), K 0 (π E , ψ)) is equal to K(r l (π F) (l) , ψ l), as representations of P n (F), where P n (F) is the mirabolic subgroup of GL n (F). Now, to prove the main theorem it is enough to check the compatibility of the action of π E (w) on the space H 0 (Gal(E/F), K 0 (π E , ψ)) with the action of r l (π F) (l) on K(r l (π F) (l) , ψ l). Such a verification, as is well known, involves arithmetic properties of the representations. We now assume the hypothesis of theorem (1.1). To complete the proof we show that it is enough to prove that