Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 93-106.

ON THE NON-VANISHING OF SHALIKA NEWVECTORS AT THE IDENTITY

Harald Grobner and Nadir Matringe

IMJ-PRG, Université Paris-Cité, 8 Pl. Aurélie Nemours, 75013 Paris, France
e-mail: nadir.matringe@imj-prg.fr

Abstract.   Let π be an irreducible admissible unitary ψ-generic representation of the non-archimedean general linear group GL_2n(F), which admits an (η, psi;)-Shalika model S_ψ^η(π). In this paper, we show the nonvanishing of all non-zero Shalika newvectors S^o ∈ S_ψ^η(π) at the identity matrix g = id ∈ GL_2n(F), if η is unramified. This complements the analogous result for Whittaker newvectors, which can be read off the formulae established independently by Miyauchi in [Miy14] and the second named author in [Mat13].

2020 Mathematics Subject Classification.   11F66, 11F70, 11F85, 22D10.

Key words and phrases.   Shalika models, Whittaker models, non-vanishing, local fields.

DOI: https://doi.org/10.21857/ydkx2cv259

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