Finite period vectors and Gauss sums (Q6565502)

In this paper, the author studies four families of sums associated to irreducible cuspidal representations of general linear groups over finite fields, namely, the Jacquet-Piatetski-Shapiro-Shalika, Flicker, Bump-Friedberg, and Jacquet-Shalika sums. For the last three newly explored cases, the corresponding gamma factors are expressed explicitly in terms of Gauss sums. The Asai and Bump-Friedberg gamma factors over finite fields are related to the ones over non-Archimedean local fields via the construction of depth zero supercuspidal representations. It is shown that the exterior square and Bump-Friedberg gamma factors agree with the corresponding Artin gamma factors via local Langlands correspondence and Deligne-Kazhdan close field theory. Finally, the author examines the four families of period and vectors, and their connections to the corresponding gamma factors.