Tate cohomology of Whittaker lattices and base change of generic representations of \rm GL_n

2022/04/05 by Dhar, Sabyasachi, Nadimpalli, Santosh
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2204.02131

Abstract

Let p and l be 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 a generic mod-l representation of \rm GL_n(F) and let π_F be an l-adic lift of π_F. Let \mathbbW⁰(π_E, ψ_E) be the integral Whittaker model of π_E, i.e., the lattice of ℤ_l-valued functions in the Whittaker model of π_E. Assuming that l does not divide |\rm GL_n-1(\mathbbF_q)|, we prove that the Frobenius twist of π_F is a G_n(F) sub-quotient of the Tate cohomology group \widehatH⁰(\rm Gal(E/F), \mathbbW⁰(π_E, ψ_E)).

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